What do you mean by "tables with buy-ins as low as $5"? If you are talking about cash games, then a table with a maximum buyin of $5 and blinds of 2c/5c is normally called 5NL. 5NL is indeed very low stakes, in fact the second lowest available on most sites. And if you feel ready for 10NL, thats fine. I will return to the question of
bankroll management later.
If you are talking about tournament with a buyin of $5 (or more typically $5.5), then the stakes are not quite as low as a 5NL cash table, because fairly significant money can be at stake, if you run deep. I recently won a $5.5 tournament for a profit of $670, so towards the end of the tournament I was making decisions for 100´s of dollars, which will not happen at a 5NL cash table.
My second question is, why do you want to move up? If its for more excitement, especially with tournaments, where more expensive ones allow you to win more significant prices, then you should make a poker budget and stick to it. If you spend the budget for the month, you stop playing or move back down to cheaper games.
If its for profit, then you are likely getting ahead of yourself. "A few month" is not enough to establish a track record as a winning player. So you are basically just hoping, that you are winning, or you are thinking so, because you ran hot over a small sample. It can however be ok to basically skip 2NL and 5NL and go to 10NL very fast, if you are playing cash games. Then you just need to give yourself a reasonable starting bankroll like $500 and move down, if you drop below $300 (30 BIs).
With tournaments though I would not suggest a new player to start any higher than $5.5 games. And this also require at least a $500 starting bankroll. So rather than trying to force the issue, just begin with such an amount and then let your results determine, when you move up or down. I personally use a 100 BI rule for tournaments, which is simple and has worked well for me for years.
Right now I have around $2.600 in my PokerStars account, so the most expensive tournaments, I will play, are those with a buyin of $22, or for SnGs $25. And I will also mix up buyins. In the session, I am finishing right now, I only played one $22 MTT and one $25 SnG. The rest of my games ranged from $5 to $16.5, which prevent me from losing a huge chunk of my bankroll in a single session.
As for what to expect from more expensive games, then obviously the average skill level will increase with stakes. In cash games the largest gaps tend to be from 2NL to 5NL and from 10NL to 25NL. With tournaments its a bit more complicated, and it will also matter, when you play, and what kind of tournament.
Right now I am on the final table of a $11 PKO, which began at 5.30 am CET. Most of the people making the final table were from Canada or South America, so they are playing late at night, some of them likely drunk as well, since it was a saturday night for them. And as a result this final table is much softer, than what you often see in $2.2 - $4.4 tournaments played at other times.
The way to adjust to better opponents is to move closer to a GTO playing style, if you dont have particular reads on them. If they 3-bet you, then defend with the range, you are supposed to defend with, and so on and so forth. Your goal against good players is to lose as little as possible. The goal is not to beat them or prove, who is boss.
Your profit is always going to come from bad players. Which also mean, and unless a more expensive tournament still have at least some bad players, then maybe you are actually better off playing the cheaper one. Regardless of buyin you want to avoid playing in games, where you are only battling it out with other good players AKA regulars. And this is true for both cash games and tournaments.
