You should start to think when there is a "contact" between your pieces and your opponent's. Don't spend too much time in the opening. Develop your pieces quickly
How to Play Better in Quick Games

I'm by no means knowledgeable in this, but I'd imagine memorizing opening theory would greatly benefit blitz and bullet players, since that means your first 8-10 moves come out rapidly, while being sound. At very short time controls, that could be a huge advantage. 10 moves is no laughing matter in that type of game.

First of all, like others said, a 1600 online rating is approximately 1200 blitz. Give or take a hundred, but really this kind of gap is normal.
As for finding moves fast, I'm not sure how to get better at it other than playing more blitz. This wont improve your chess overall, but it will help you play things you know faster. In my opinion it's a lot like knowing multiplication tables. If someone says 5x5 you don't get out a pencil and paper or count on your fingers, the number 25 just pops into your head. The better the blitz player, the more tactics and threats are immediately obvious to them after every move.
The in between times where there aren't tactics you use things like strategic knowledge and endgame knowledge to find good candidate moves quickly. To improve these areas (as well as the tactics) blitz play alone is not enough. As I said above blitz only makes you faster at what you already know. If you don't know it to begin with you'll just be making random moves.

I just don't see the whole board as well during blitz games. I don't feel I have time to.
Yep. The better the player the more they're aware of instantly.
Perhaps a better example is doing math in your head. A child can't, because every step is difficult and they're unfamiliar with certain patterns. However after years of practice and experience ask the same person a complex problem and they may be able to run some numbers in their head and give you a very good approximation without solving the problem explicitly.
A simple example may be what's 7/25? The way I would do this is to note this is the same as 7/5 then /5 again. 7/5 = 1.4 because I know each 1/5 = 0.2 then divide 1.4 by 5 is a little less than 0.3 because I know 5x3 = 15. And a very quick mental calculation gives 0.28 which is the answer.
That's a lot like blitz chess I think. I didn't do any actual calculation to get the approximation "a little less than 0.3," it was all patterns. Which is like blitz chess. There's no time to calculate and analyze specific solutions, so people are using their experience and patterns to get approximate solutions quickly. The more you practice this, the faster you'll be... but you wont necessarily be better at math or chess in general... just faster at making approximations.

I just realized there's a much faster way too... 7/25 is the same ratio as 28/100 (just multiply by 4) and poof, instant answer -> 0.28
This faster method which got the answer with no calculation at all would correlate to the better blitz player, but not the better chess player / mathematician!

So then what do I have to do to be better at games of 30 minutes of less? I'm only fifteen years old, so yes, I have a lot of chess ahead of me, but surely there's something other than just gaining experience. I feel that I am a bit weird at my age in terms of chess because I prefer long games. I like to have an hour- or even more- per side in my games. I'm still not a good OTB player- my USCF rating is about 1050 in standard time controls- but I think a lot of other players at my age are much more comfortable with quicker games.

So then what do I have to do to be better at games of 30 minutes of less? I'm only fifteen years old, so yes, I have a lot of chess ahead of me, but surely there's something other than just gaining experience. I feel that I am a bit weird at my age in terms of chess because I prefer long games. I like to have an hour- or even more- per side in my games. I'm still not a good OTB player- my USCF rating is about 1050 in standard time controls- but I think a lot of other players at my age are much more comfortable with quicker games.
There are two things that can hold you back. First is just speed. To get faster just play lots of blitz games. You're going to feel confused and you'll blunder, lose, and it will be frustrating. As with many new skills as you continue to fail and get frustrated you'll develop strategies to succeed. Many lower level blitz players just make random threats over and over waiting for their opponent to miss one. Some try to keep everything defended and shuffle pieces back and forth so their opponent will lose on time.
The second thing that can hold you back is not having enough chess knowledge in general. In this case don't play blitz at all. Play long games and study.
Really good blitz players have alternated between these. They've had long periods of study with no blitz, as well as lots of blitz experience.
My advice would be the 2nd piece... ignore blitz for now. So what if other kids like it, if you don't then don't play it! In the long run you'll be a better player by avoiding blitz and when you do decide to play tons of blitz you'll have a better foundation to streamline. For example instead of incessant 1 move threats or waiting for them to lose on time you'll be able to attack and defend and follow the strategic theme of the game.
Lots of blitz in the beginning has a downside too. The strategies you develop to help speed yourself up can involve bad habits. Assuming recaptures is an easy one to point out, if you let this become a habit it will make serious analysis during long games harder. Not that you'll be doomed to be a weak player if you play blitz now, but it can set you up for roadblocks in the future.
But again, if you want to be better at blitz in the short term, then play lots of blitz and nothing but blitz (with some tactic puzzles on the side, but mostly just playing).

You aren't bad at blitz. Are 20 US dollars worth less than 30 Japanese yen? No. Is a 1250 blitz rating representative of a lower standard than a 1600 online rating? I don't think so. I would say that 400 is a fairly average difference between the two.
As for improving, well I'm no expert but my intuition tells me that experience/knowledge is more important in faster games than longer ones (as samir approximately said). You want less calculating and more reflex.
my blitz is higher than my online. which probably means dpnorman really is a bad blitz player
Your ratings are bizarre, a big exception. I'd guess you don't take your online games very seriously at all. Maybe you're letting a friend play the online games while you're the one who plays blitz?

I honestly think that's cool that you prefer long time controls at only the age of 15. Btw.. How long have you actually been playing chess? The only reason I ask that is because a lot of it is time and experience. Time is going to pass whether or not you want it to, and it's up to you to decide how much experience you'd like to put under your belt. Also, studying the overall game can be a tedious task, but it will benefit you in long games, and blitz as well.
Now, I just thought I'd throw this out there, but blitz comes down to a lot of non-sense. Everybody is liable to make a blunder while playing blitz (as i've seen many masters and GM's do on this website countless times). My point really, is use blitz as a tool to try out new ideas. Get creative w/ it, and have fun. Also, I don't know if you have any openings/defenses that you play on a fairly regular basis, but it definately helps in blitz when you've played a game of chess you've played a thousand times.
I'm not sure how helpful this was, or how closely related it was to your original question, but I thought I'd give my two-cents.
Oh...! I almost forgot. When you're playing your blitz games, try to learn to trust your intuition. In your every day life (especially as you get older) you'll be making a lot of choices based on your intuition (from experience, logic, etc). Intuition in chess comes from hard-work, study, confidence, and belief in your ideas.
Good luck.

You aren't bad at blitz. Are 20 US dollars worth less than 30 Japanese yen? No. Is a 1250 blitz rating representative of a lower standard than a 1600 online rating? I don't think so. I would say that 400 is a fairly average difference between the two.
As for improving, well I'm no expert but my intuition tells me that experience/knowledge is more important in faster games than longer ones (as samir approximately said). You want less calculating and more reflex.
my blitz is higher than my online. which probably means dpnorman really is a bad blitz player
Your ratings are bizarre, a big exception. I'd guess you don't take your online games very seriously at all. Maybe you're letting a friend play the online games while you're the one who plays blitz?
look at your own ratings. your blitz is 400 higher than your online
Ratings that aren't established don't really count, what have I played, 5 maybe? Also these games were against people on the forum who wanted to do casual / fun / practice type games.
On my first account "orangehonda" my online rating was 2100 after only 10-20 games.

Very interesting. Maybe the ratings have changed since I was active here. In the past online ratings were much higher than all of the live ratings (blitz, bullet, standard).
One thing to be careful of though is those who have just a few games or very old ratings, those ratings aren't as accurate.
Hey guys. I play a lot of online chess and 30 minute live chess and I have been having issues lately finding good moves quickly. I feel that even 30 minutes is not a long time for me and in my 30 minute games I frequently find myself in time trouble. I also play blitz occasionally, but it's a rare occurrance for me to finish a blitz game without a blunder; my blitz rating is three or four hundred points behind my online rating. I want to know how I can become better at playing in quick games- I don't think my problem is tactics but rather just finding good moves quickly.