In general, try games until you find one that compels you to get better at it without forcing it. Shmups are not for everyone, you'll make a lot better progress if you enjoy a game rather than banging your head against the "top ten" list of a bunch of nerds. You should get mad when you die and instantly want to start over.
That being said, try a couple things. First, pick a caravan mode or two, like Final Soldier or Soldier Blade, and play the 5 minute mode. Try to figure everything out about the scoring you can. Because these games are so short, you can really max them out with a lot less effort compared to longer games. These will teach you a lot of the skills you need to learn to get better at improving on your own. Second, try some older games like Star Force, R-Type, or Twin Cobra. These games are all enjoyable and can be difficult to start with, but you might find you enjoy this era of games more than some modern stuff. Next, try something like Giga Wing, Dodonpachi, or Espgaluda. These are accessible enough for beginners while still having depth and are a bit more modern. Others are probably more familiar with newer games than me and could recommend some stuff. If you want to get crazy, maybe try Perfect Cherry Blossom. All in all, skip around until you find what you like: hori vs. vert, what companies' styles, what era, etc. Darius II might be a good place to start in the monthly shmup thread here, as well. Ask for some strategies or bring questions to the thread, there's lots of helpful people here.
