Now I know that steampunk is generally done in around/past the industrial revolution (and that what I'm lookong to do may not technically even be steampunk), and for good reason. However I'm looking to build a world where a great calamity has struck an advanced civilization, throwing its people back into an era more akin to the medieval times. That being said, traces if this civilization still exist, and people have both learned from it and used the parts to build their own technology. The question is what would that technology look like? I'm thinking still early in the dark era, so they still wouldn't understand most of it and would not have the tools to create any of the finer pieces. But how would the presence of this ancient technology affect the tech growth of the current people?
You are right that this isn't steampunk, it's certainly an interesting world-building possibility though.
This is something that has been looked at numerous times throughout fiction and RPGs so you can certainly find elements for inspiration in all sorts of places. For example Numenera, Babylon 5, Against a Dark Background, even Fallout.
At least initially the famous Arthur C. Clarke quote would apply "Any sufficiently advanced technology is indistinguishable from magic".
The wizards of the time would use the scraps of technology without understand. The mystic incantation "ok google" that activates the demon in the box. Some technology would reveal itself more easily, anything mechanical in particular can at least be understood if not duplicated.
The existence of scientific records, or scraps of them, would allow faster progress in many areas - but so much of what is described would be meaningless without the context to back it up.
I think Asimov's Foundation "trilogy" might give a rough idea of this:
In that series, as humanity declines back to the dark ages on hundreds of worlds, the operation of advanced technology becomes like a religion-- people know how to use technology, not how the technology works.
For example, they might know that they can light up all the buildings in a city by performing a certain ritual before the "Djinn Erraytor"...but they don't know that pressing a certain button primes a pump, or adjusting a dial until a needle falls within a certain threshold will set control-rod depth and to prevent a meltdown.
We have a real-life analogy for this: The fall of Rome. People lost the ability to build things like the Romans had built, but they could see the decaying artifacts all around them. Monks preserved the old knowledge until people were able to rebuild the economic base to use it again.
The result was that people had great respect for the abilities of the ancient Romans for centuries afterward. Even after they surpassed the Romans they still seemed to have an exagerrated respect for them. But no one seems to have thought their technology was magic or incomprehensible. There was a very sensible attitude of, "We need to study these ancient books and learn to reproduce what they did, and figure out how they were able to do things that we can't."
Enclaves could maintain some technology. Like, I think a small group could build a hydroelectric generator to produce electricity, make light bulbs and other simple electrical devices, and maintain a 19th century technology. But some things require a huge technological base. I have a hard time imagining a few dozen or a few hundred people getting together, reading old books, and building a nuclear power plant or a microchip manufacturing plant. Some things just require too many specialists with diverse knowledge, and too many components that are presently made by different people all over the world.
Clearly something like "building a cell phone network" requires the involvement of very large numbers of people over a large area. Even if you they are able to build the phones and towers themselves, who's going to build and maintain all these towers, and all the lines to connect them?
Some technology could be kept running but would not be replaceable when it broke down. Like if someone did make a small electric generator, he could plug in a computer and use it, but once that computer broke, he'd be unlikely to be able to repair it. Like the old Roman roads and bridges, they could be used until they wore out or collapsed. Or like the Colliseum ended up being used as a zoo and finally a garbage dump.
As people rebuilt, progress from medieval technology to modern technology would be much faster than it was the first time, because they don't have to reinvent everything. They have the old records to give them a shortcut.
You might want to read "1632" and" "1633", the first books in a time travel series, as your question is very thoroughly explored by the authors, and ditto with the associated "Grantville Gazette", at least the first volume if not others.
Basically, there are two variables here:
How long is the period between the fall, and when a stable enough society (even a small one) emerges from the chaos, with the ability and the will to rebuild?
Is the interregnum long enough for all books to have vanished, or are they still around?
It's been pointed out that all you need is a set of the Encyclopedia Britannica circa 1880, and you can reconstruct steam-era tech. That should be self-sustaining even with a relatively small group... say, a small city or several cooperating villages in close proximity, with a total population of 2000 or more.
Electricity, as Jay suggested? Well, if you can make wire -- and heck, you can do that even with medieval tech -- and have the instructions for making an electric motor or generator, then certainly they can generate electricity. But what are they gonna do with it? Making light bulbs is much, much more difficult, and mass producing them is a very specialized industry that isn't likely to arise very soon. Both mass producing light bulbs, and building an electrical distribution system to send power to factories and homes, both imply a highly organized, large society... which isn't going to emerge for some time, given your scenario.
If it was me, I'd go with Watt (reciprocating beam) steam engines (not later, more efficient and more advanced steam engines) used as stationary power generators for industry, and gaslights. Since you used the term "steampunk" in your question, hopefully that will fit right into what you're looking for. And keep in mind that there were steam engines around a long time before railroads, which evolved from tramways using wooden rails; and it was a long time before steam engines could propel ships across the ocean. Think early steam age, not late. Think canals and tramways with horses or mules as the motive power, not railroads. That tech would be a lot easier to build and maintain without a large population base, and without thousands of mechanically skilled, trained workers.
The biggest obstacle to rapidly advancing tech is going to be lack of machine tools. When Watt made his first steam engine, the state of the art of boring out a metal cylinder was so poor that he could fit a shilling between the piston and cylinder! You need the tools to make the tools to make the tools... and they just don't have the tools to make precision metal-working tools. Most or all of those will rust away long before the books turn to dust. Even with instructions for making advanced, more efficient steam engines -- or even more advanced machines -- they won't be able to make them. Significant advancement towards a higher tech level would require an industrial base formed from a sizable, and at least semi-educated, population base. Something on the order of at least hundreds of thousands, and more likely millions, all working under the same government.
I think a lot of different factors are to be taken into account here.
Do they speak a similar language ?
There are printed versions of Wikipedia. Books, encyclopedias, anything. If there is leftover tech, there is leftover books, and those would be huge.
Note that there is no need for a lot of books: the first group of men to control a university's library will just rule the surroundings, able to get back to a perfectly normal level in less than a century (total guess from me though)
So, next, I will imagine that all libraries were lost and/or that no one knows how to read. Eventually though, that knowledge will be back.
The technological difference
A caveman, seeing a car, will not think "I could drive that". At best, it would use it as a house. Depending on the technological regression, that kind of thing could happen for most objects.
If the difference is small, they will be able to retro-engineer stuff and science will get a big boost. If not, technological objects will be seen as "mostly the materials they contain". Metal is always valuable, and if salvaging stuff on the ground costs less than digging down for it, salvaging will happen.
Of course, a minority will say that those objects should be studied or that "it belongs in a museum", but most people will be busy making their immediate life sustainable in a post-apocalyptic mayhem.
The maintenance problem
Most technological devices require maintenance and/or an energy source. Technology doesn't work just because it's here. It works because we're here to make it work and we know how. Producing batteries, operating powerplants, repairing broken stuff, that's impossible to do without knowledge.
This would probably add to the fact that they would not understand anything about the technology they're handling: imagine you don't know what a car is, you don't have a the car key, or gas, or a new battery and the car's computer system is down. From the wheels, you can understand that it's a vehicle, and where the driver should sit, but there is no way you can start the car, or understand the engine. Most likely, you would look for a way to put a harness on it.
So would it have an impact at all?
My guess is, most people would fight for the buildings and materials, without trying to understand the tech itself.
People in power though, would keep stuff intact and pretend to understand it for fame. And a man influential enough would pay for research until they manage to find books and read it.
Well, the internet was supposed to survive that sort of calamity. Let's just say that somehow the transmission lines for the net remained intact, and that at this advanced stage of civilization the power for the internet had somehow been taken off any notion of a grid, and ran on like a source of perpetual energy, or something mooched from the environment -- in that case, even damage to most of the nervous system of the net would still leave operating circuits. Most probably, at such an advanced stage civilization, devices something like iPhones were widespread, and the notion of IoT and interoperability of component parts also so, so there could be a diverse array of "found" devices, which can still plug into the remnants of the great global web. Of course...hackers, bitcoin, BBS, other things can still exist because of this information backbone that remains. If you consider that, in essence, life is an "information technology" and it is something capable of persisting by being self-sustaining with respect to resources in the environment, I think a slightly more developed iteration of our internet, which can somehow become self-sustaining off the resources of our environment, is the most likely type of technology to survive myriad calamities across the greatest number of possible worlds.
For the first while (perhaps several centuries past the apocalypse world), you would not have a "steampunk" universe, but rather a "scavengerpunk" universe.
In previous "dark ages", people may have thought of the structures and temples of the past as mysterious relics (the Ancient Greeks evidently thought the Palace structure of the Mycenaean civilization as being built by the Cyclops), but generally "free game" to be repurposed as building materials for walls, rubble for road building or metal parts melted down for more immediate uses.
In your more enlightened age, people will be scavenging cities for abundant metals, and dismantling machinery for usable parts. There could be a whole "hillbilly" tech of repurposing parts to keep old cars and trucks going, pulling apart devices for pipes to make stills (for alcohol fuel and "medicinal" purposes), presses for reloading firearms ammunition and so on. Electronics will be much more difficult without the infrastructure for testing and manufacturing, and many electronic devices will be "killed" because there is no "clean" source of electrical energy (a voltage spike from a defective windmill or bicycle generator will do in a computer, for example). The movies "Mad Max" and "The Road Warrior" show a society degenerating to these levels as oil becomes scarce to non-existent, so this is a good place to guide your thinking.
Several answers have suggested that books will be important, which is true to a certain extent, but unless you have salvaged detailed technical manuals, it will be difficult to keep things in running order. Don't forget that the infrastructure to make all the parts and tools is gone, so until you either scavenge the part, repurpose something different to work or make the tools to make the tools to build the part, you could be stuck with dead pieces of machinery, engines etc.
In the end, people will be heavily influenced by the mechanical artefacts from the past age, simply because they are understandable and can be passed down through the years with maintenance and close observation. Electronics, especially solid state electronics, will become a lost art, simply because there will be no way to repair, maintain or make new examples, and electronics will have to be rediscovered at a later date.