Remember that a bet should have a specific goal in mind, e.g. to isolate, to get multiple callers, to steal, etc.
In most cases though, we want to balance/disguise our range between premiums and air by making it hard to tell which we have. IOW, you're almost always representing some strength with your preflop bet, whether you actually have it or not, so you want to bet air the same way you bet the nuts. If villain knows we tend to bet X with air and Y with the nuts, then it's an easy game for him. I think we touched on this in your Position thread, but as was said there and here, don't alter your bet size based on the strength of your hand. That is highly exploitable, you're basically playing every hand face up that way.
Here are some common factors that will cause you adjust your pre-flop bet sizing:
- In relation to the blind levels. This mostly applies to tourneys. As blinds increase in relation to stack sizes, then opening raise sizes should decrease so that you're not over-committing your stack on your less premium hands, at least until you're in push/fold mode. You want to be able to play your good hands like mid PP's or maybe suited broadways, but still get away from them when it's clear you're beat. How much you adjust sizing will vary by person, but as a general rule I'll start a normal tournament with a standard 3x open when stacks are deep (say at 10/20 or 15/30 blinds). Then when the blinds get to 50/100, I'm opening 2.5x. Then somewhere around 100/200 maybe I go to 2.25x, and then at 150/300 or 200/400 and beyond, I'm 2x minraising until I'm short enough I have to shove or fold. Another advantage here is later in the tourney, when chips are more valuable, your premium hands will be more likely to get action because you're betting smaller.
- In relation to position. This applies mostly in cash games, but sometimes tourneys too. Some people bet more in early position and less in late position, for example 3.5x UTG, 2.5x OTB, and 3x in between. Or some will add 1bb UTG/+1 and every other position be the same. Reason being, in early position if you bet too little you risk getting a lot of callers, which generally isn't what you want especially with PP's. It helps to slightly reduce your positional disadvantage. In late position you don't need to bet as much to limit the field if only the blinds behind you might call, plus you have positional advantage, so you risk a little less before you can see a flop. You can always build the pot later on a favorable flop.
- In relation to the number of players in the pot. When you want to either isolate/narrow the field with a value bet, or punish limpers by stealing their weak bets with ATC, you'll start with your standard open (say 3x) and add 1bb for every limper in front of you. This is to price out callers behind (or narrow their range) and shake loose the weaker limpers, depending on how many you want to the flop. Normally with your bigger PP's you want to take the flop HU or 3-way, while with your speculative hands like suited connectors, you prefer a multi-way flop for the implied odds (in which case you're usually just calling).
- To induce a certain reaction based on a read. This is a bit advanced and requires good reading skills and paying attention to other players involved. It's one of the only times your hand strength might influence your bet size. If you've noticed a villain reacts to certain bet sizes or patterns, you can exploit that. For instance, if a villain always shoves on overbets because he considers them bluffy, then overbet your strong hands in order to induce the shove. Or if you're a small stack and big stack bully is running over the table and 3betting every minraise, then minraise your big hands. There are a number of similar ways to manipulate predictable opponents with your bet sizing. Again, this works best when you've already narrowed the field to a HU situation -- be aware of who else has to act, because you don't want to invite the wrong people into your craftily calculated trap. If you're not careful here though, you'll wind up trapping yourself, so don't overdo it. Save this tactic for when your game has advanced a bit.
- To apply pressure when you're the table captain. Like bubble situations where you're the chip leader with a huge relative stack, and a bunch of short stacks are sweating the bubble. You might normally be minraising at this point, but with your safe stack you can increase your bet sizing to force the shorties to commit their entire stacks if they want to play. Being the bully is very effective, as long as you can recognize when to back down. You have to be careful though because if you do this too much with air, you might price yourself into having to call their shove and risk doubling up a shortie, which can be dangerous even though the chip loss didn't really hurt you.
There are other cases that fall in between some of those, like squeezing for instance. But barring a specific situation where you're trying to achieve something out of the ordinary, you should NOT be sizing your bets based on hand strength.
As to a 10x opening size, that would be hugely unconventional online where only the most clueless players tend to do this. In live games, however, opening sizes can run the gamut, and I've seen some crazy games where 5-10x was standard, and you have to adopt (and adapt to) that standard.