1) Jump over to Perfect Cherry Blossom and start with that game first. The bullet patterns aren't as random, you can see your hitbox while focusing, and hitboxes are actually smaller on all character than in EoSD (Reimu has the smallest hitbox of the whole series in this game, along with extra time to death bomb). The game's gimmick is also a one-time shield that you get about 3-5 times per stage, which allows you to make more mistakes while learning. Imperishable Night is also a very good game to start with difficulty-wise, but all the different mechanics (Time orbs, yokai/human gauge, etc.) can be overwhelming for a newcomer.
2) Use your bombs any time you remotely think you're going to be hit. Bombs are use-or-lose in Touhou (i.e. they don't stack on death), so it's better to use a single bomb when you weren't going to get hit than actually getting hit and losing three bombs in return.
3) Playing Touhou is like driving a car, you need to anticipate what's coming instead of only looking at what's right in front of you. Start shifting your focus to halfway up the screen to see how where and in what directions bullets will be going. This usually gives you an extra second to two to think on what the best route to dodge bullets are, or if you'll need to use a bomb or not. The death bomb window is too small to do it on reactions alone (bar PCB/IN Reimu), so you bullets to use it properly.
4) Download the
Touhou Practice Application and use it to practice sections that are giving you a hard time. This tool instantly lets you jump to sections of stages and any part of a boss fight, which is leagues better than the in-game practice mode. You can open nearly all the games through the application itself, but EoSD requires that you place of a copy of the thprac.exe file in the game's folder and run it in Japanese (done through
Locale Emulator).
5) Just play more. I thought I was going blind when I first started playing the Touhou games, but your brain slowly starts adjusting to the amount of clutter on screen the more you play it. It's hard for me to describe, but it's one of those things that just comes with time and practice.