Honestly, as long as the populations of the world is brought down to a small, SMALL percentage (even something as large as 1% could be fine), you should be okay. (I chose 1% because I believe that is the largest percent of a group that could survive for a true apocalypse, but that's just my personal view. I'll operate under this idea. You can modify it as you see fit.) We're at 7.6 billion. 1% of that would be 76 million people. Sounds like a lot, yeah? Well China's population would be about 13,860,000 or so of that. India's would be another 13,390,000 or so of that. That's already over a third of the remaining population in EXTREMELY large countries.
If we were being generous, that leaves the rest of the world with about 50 million people. Let's just assume all 50 million people SOMEHOW lived in the United States. (Not even Canada and Mexico, just the US.) That's 15% of the current US population. Mind you, this means the only places in the world with surviving human populations are USA, China, and India. Nowhere else in the world is there another survivor. Seems pretty unlikely that the population distribution would be so densely focused. So, let's spread this out a LITTLE more.
If we were to include Canada and Mexico, this means North America's population just dropped to 10 percent of what it was. The last time the US Population was at 50 Million was in 1880 when the US was in the process of growing at an extremely fast rate. If we assume all other humans exist in only China, India, Mexico, and Canada (the latter two would likely try and move towards the US or the people of these nations would attempt to work together with the US to some extent), then that means there's not going to be an influx of immigrants. What the US has is what they have. If we assume the growth rate of the new America mirrors that of the world today (a 1.09% yearly population increase), then it would take about 175 years for the population to mirror that of the US today. It would take a little over 200 years for the population of the new America to mirror that of the combined US, Canada, and Mexico today. This is also assuming that America somehow managed to be practically unfazed by the apocalypse in comparison to the entire world, even compared to the only two other surviving regions.
In reality, assuming a 99% decrease in population, the US's population would drop to around 3,257,000. That's significantly less than the population of Los Angeles and less than half the population of New York City. Canada would drop to around 367,100. Mexico would drop to around 1,292,000.
What this means is if you go out to a mall or a crowded area, of the people that you would see now, only a handful of them at best would still be around. Odds are, everyone you know and trust will have died and you now have to survive on your own or with the help of strangers. Seems fine, right?
Well, as seen in the answer here, power would stop within a few months if you're lucky. In reality, it'd probably shut off in a few hours without someone manning the controls and keeping things regulated and even. As for the water, according to Life After People, a fictional apocalypse documentary, water would last maybe a few months. If you have a well and your own alternative source of power, you could get that to last longer, but that'd be rare exceptions for someone prepared for this. It would not be the norm. Internet would go out pretty quickly as it would not be a priority. Radio would last a long time due to a lot of stations having backup generators, but even then it'd go out sooner or later.
On the upside, you'd have your pick of housing. With 1% of the world's population left alive, this means, if we assume EVERY house had a mother, father, son, and daughter, then only 1 out of 25 houses would still have a singular resident. You really could take your pick of new abode. If we assume the deaths/disappearance/etc. of people was truly random, not taking into account age, health, and so forth, this also means that there is a good chance that the house you want could be someone too old, too young, too weak, or too sick to be able to defend the house. Odds are this wouldn't be how it actually worked out, but if we were going with raw probability of survivor, it works for the sake of thought.
Without the internet, most people wouldn't know what to do. There are three main things to consider when initially trying to survive and those are Food (and, by extension, Water), Clothing, and Shelter. Well, housing is readily available, so that won't be an issue. You can always raid the closets of the houses around you for clothing if you didn't have enough of your own. All that's left is food (and water). If you act fast, you could raid water jugs and bottles from stores, run them under the tap, and fill up enough water to last you a long time. A half gallon (really 64 ounces) of water is your daily recommended intake. You could probably get away with half of that (32 fl.oz. or a quarter of a gallon) if you want to conserve resources or a quarter of that (16 fl.oz. or about an eighth of a gallon) if you are relatively inactive and sedentary, maybe. Mayo Clinic actually recommends nearly double the usual 8 cups of 8 ounces a day, meaning while you may be able to get away with less, you probably shouldn't. Especially not if you're not eating appropriately. As the WebMD source says and the earlier Mayo Clinic confirms, about 20% of our fluid intake actually comes from other sources, namely food. This is the simple part. I'm pretty sure everyone knows to raid the pantries of nearby homes to gather food and drink. That's the easy part...
Then comes the hard part. Most people who lack experience in agriculture or military backgrounds will likely be unsure of how to start and maintain a thriving garden to live off of, let alone a farm. In fact, most people will likely see their storehouses of water and think they're fine where they are. Unless these people live near a source of clean freshwater, they shouldn't settle where they are already at. Even if they do live near a freshwater source, there's a chance it may not be clean enough to risk (especially in countries lacking in the same dumping regulations as the US.) This means anybody who isn't constantly moving towards cleaner sources of water is actively put at risk of dying from dehydration once their water runs out, sickness from the contaminated water, or dehydration caused by becoming sick thanks to the contaminated water. Once they get there, if there aren't people used to farming, fishing, or agriculture, they will quickly run out of food and you'll get raiders who are hungry and desperate stealing and killing for food. You may even wind up with worse atrocities like (essentially) slavery and other things depending on the morality of the survivors and how desperate they are for the things they want. You may get tribes of survivors starting to form together, but even then, it'd be a challenge. I honestly see dying out as a more likely scenario, at least in the US and Canada, either due to in-fighting, starvation, or lack of people to procreate with. In more rural countries, a new civilization COULD occur, if they can find other people, but again, they'd have to find people who are willing to cooperate and work together, sharing their skills and resources with others.
No matter what, within 10 years, you can expect petroleum products like gas to run out or be unusable, so don't expect people to fight over it... It won't matter. Honestly, it'd just be a day to day grind where humans inevitably die.
If animals and plants are affected by this too, then this will speed up the process for humans to die off. Insects and small animals will be able to resurge thanks to how fast plantlife regrows, but most larger animals will struggle as they fight over food. Wolf packs will have been disbanded and erased. Flocks of birds would be down to a couple members. Only patches of grass will survive. Most trees will have fallen. The world would regrow, but it would take a long time for animal life to make a comeback and humans will struggle to hunt and forage in this new world. You could expect this world to still be livable, but it's a matter of if humans will succeed. Once humans do manage to lay down roots again, though, expect there to be a resurgence in that respect too.
I'd say, post-apocalypse, 1% of the world (76,000,000) will remain. If I were to make a wild assumption, 1% of the survivors (760,000) will ultimately survive the post-apocalypse. Assuming a 1.09% growth rate (the current growth rate of the world population), humans would reach 1 billion in 663 years, the same approximate population of the world in 1800. Odds are, by this point humanity would be back to a similar way of life as in those days. Without regularly inducing apocalypse now-ish, you can't really expect the world the stay post-apocalyptic, especially if you don't want permanent environmental changes, which would happen no matter what form of apocalypse were to occur.
Basically, you're looking at turning the clock back on civilization by about a thousand years. It gets the job done once, but that doesn't assure it will work the second, third, or fourth times. With each event, humans will become more resilient and prepared, with bunkers already prepped and stocked for the next time and a larger emphasis on physical record-keeping which is likely what kept the humans from the first apocalypse alive: those who were smart enough to hit the library.