Use a few scales
The first and biggest is the Kardashev scale. This measures the total amount of energy the civilizations has harnessed on a logarithmic level.
K-1 is "has harnessed all of the energy the sun would deposit on Earth".
K-2 is "has harnessed all of the energy the sun emits, period".
K-3 is "has harnessed all of the energy an entire galaxy of stars emits".
Each 0.1 on the Kardashev scale is a factor of 10 -- these steps are $10^{10}$ or a factor of 10,000,000,000x apart from each other.
Human civilization is currently around 0.72 on this scale, give or take a few 0.001.
With this scale you can measure "can they dismantle a moon"? "Can they launch a slower than light ship to a nearby star?" - both of these are fundamentally questions about energy budgets. The engineering may be tricky, but once you can deal with energy on the scales required, they aren't fundamentally difficult to do.
But energy isn't the only thing that a civilization technology can be measured on.
Two other ones I would suggest would be information or computation and the ability to manipulate and control small things (and how small they get).
For information, just a simple logarithm of the bits of information we have catalogued, stored and can manipulate. Humanity today is in the 10^15 bits range.
For small scale control, humans can futz around with stuff far smaller than they can control.
We have nearly full control over stuff at mm and above scales.
At nm (nanometer) scales our ability to control stuff is in clean room labs with limited materials. We can print computer chips at that scale, but we have limited control over DNA, for example, or biology at that scale.
So how about 3 numbers
- Log base 10 of the Watts the civilization has access to. (Humanity is 7.2)
- Negative log base 10 of the scale of things we can manufacture (computer chips are 8.5) in meters.
- Log base 10 of the number of bits of information the civilization has access to. Humanity is currently about 15.
How do use these numbers
A human brain has $10^{11}$ neurons. If it takes 100 bits of data to describe its connections, that means a single human brain requires 1% of our current civilizations entire information storage to describe.
This would imply that human-level AI is at the edge of what we can possibly do. An Apollo-moonshot level of effort. This seems plausible.
If our civilization had $10^{20}$ bits, a human brain would be a trivial amount of information to process.
You can do the same for fine control -- at 9.5, Tailored DNA is possible; the information required to know what it does may not be around, but you can specify with DNA something has.
Being able to program DNA and make it do what you want is going to require 9.5 or better fine control, and enough information capacity to understand biology. There are a number of people who have worked out how much information there is to understand evolution; a single human has about 6 billion bits of information (9.5), not that high.
But decoding what it DOES could be harder. The information required to understand the evolutionary history of humanity, at ~100 generations per 1000 years, $10^{8}$ or so times larger; closer to $10^{18}$ bits to fully model the evolutionary history of humanity back to single celled organisms.
If that is what is required to create a custom species (make life do what you want without having to experiment), then a $10^{20}$ bit civilization would find custom-creating a lifeform to be an Apollo-scale task (requiring 1% of its entire civilizations information capabilities).
We don't know how hard it is to decode how biology works, but it is probably in that range.
Now, look at futurists.
Information, small scale control and energy are known ways for futurists to think about what human future capabilities look like.
So by using these 3 scales you can find futurists talking about this kind of thing and play around with what they can do.
What more, by having one civilization have advantages in one area and not another, you can have interesting differences in development that aren't "well, those aliens never noticed fire could be useful".
Growth
Exponential growth is not atypical in our technological development. On a log scale, exponential growth is linear.
So if we are at 14 on an exponential scale and get 3% better per year, $log_{10}(10^{14} * 1.03)$ is $14 + log_{10}(1.03)$ which is $14 + 0.013$. 3% growth means about 80 years to move up 1 unit on a $log_{10}$ scale.
This applies to all of information, fine control and energy.