Recommendations:
Eschatos: I'm convinced that this is the second greatest shmup ever made. The one I consider #1 is coming to Steam but isn't here yet; I have somehow not killed anything out of impatience. This game also has the benefit of the most goddamn weird final boss I've ever seen. Not that you'll get there, mind you.... but the game does give you an interesting shield mechanic to use, and two separate weapons, one of which fires straight forward in a thin line, the other fires a wave shot at angles, but NOT directly forward; you have to make use of all of these things to get through. There are multiple modes here. Like all modern shmups, the game heavily encourages scoring. It is fantastic. There's a lot of creativity here and unique mechanics.
Satazius: Like Gradius or R-Type? Classic powerups, that sort of thing? This is one that you really want. Though without R-Type's ultra-cramped levels. The level design is more like that of Gradius.
Raiden 4 Overkill: Best version of Raiden yet. You don't get much more "traditional" than this series, as far as vertical scrollers go. Lots of explosions, and the game's trademarked "seeker beam". Raiden 3 is on this list too, but that's not even close to as good as 4. This game also has a rare thing: A dual-play mode. Don't know what that is? You control both player ships with one controller, at the same time. Doesn't that sound fun?
Deathsmiles: If you want to try out a bullet-hell type shmup, but DONT want to die 2000 times in the first level, this is the one to go for. It'll wait until level 5 to do that. This is by Cave, so the quality level is high. The weird-level is also high. There's a cow, at one point. BIG cow. Throws Tetris at you.
Mushihime-sama: If you want to try a bullet-hell shmup, and you DO want to die 2000 times in the first level, buy this!
Judgement Silversword: By the guys that did Eschatos, but from a LONG time ago. I believe this was on the WonderSwan? Either way, it's a classic-style game, very good.
Steredenn: This is a scrolling shmup combined with a roguelike! You never know what it's going to throw at you, and there are ALOT of weapons. You can have your ship just outright BITE other ships. Or other crazy attacks. This is very nicely balanced, and it's clear that a lot of love and polish went into it.
GundeadliGne: No, that's not a typo, that's how it's actually spelled/written. If Touhou combined with Cave, this game is one of the results. This is actually another favorite of mine, and one of the few games that's capable of actually adjusting to your skill level. if you're not doing so well, it's not going to be as hard on you.
Gundemonium Recollection: This is the OTHER result of Touhou+Cave, by the same developer as that previous one. Both of these games are fairly easy to learn, but you do have to get used to the big sprites; they're bullet-hell though so your hitbox is about the size of 2 pixels. Another favorite of mine.
eXceed 2nd: I'll be honest, I don't care for this too much. But the reason I don't care for it too much is that it plays like Ikaruga. A lot of people like Ikaruga, so I'm recommending this anyway.
Darius Burst CS: This one is fantastic! Darius is a series that's been around for a LONG time. It's a classic-style one, with powerups and multiplayer (actually most of the games on this list have multiplayer). Darius Burst is to me the best game in the series (and the most recent). It's particularly good for those that like playing these for score, but it doesn't attempt to actually force the scoring system on you (some games will, you either use the scoring elements or you die a lot). It also has ALOT of content. Seriously. That's not something I can say for most shmups.
eXceed (the first): Okay, I take it back: Deathsmiles isn't the best recommendation for a new player in bullet-hell land. I'd forgotten about this game, you see. This is pretty good actually, but for me it was WAY too easy. Easier than Deathsmiles even, I think I took like 40 minutes with this one and had cleared it. But that means that for anyone that's not familiar with this type of game, this is a good one. The art style is a little wonky but the gameplay and such is good.
DeltaZeal/Trizeal/XIIZeal/ExZeal: These have the strangest freaking names but they're all very good. Traditional-style vertical scrollers. They can be a little focused on scoring though, and they can get a little weird about it. But these are all quality games.
Shmups Skill Test: Strangest thing on the list, it actually goes along with the Zeal games. There's a reason the image there is of a giant garbage can in space. That's something from in-game. Also, your arm will fall off partway through.
There's more to recommend but I've run out of time while typing.