http://www.shmups.com
The best console shmups site on the net I know of.
Wii Virtual Console has a lot of shmups now, that's definitely the first place to look. It has some of the greats. NES, SNES, Genesis, Turbografx, Turbo-CD...
The Gamecube just has Ikaruga and Chaos Field unless you can play imports, in which case there's also Radirgy (Radio Allergy, Rajiruji, whatever). Ikaruga is one of the best shmups ever, though, so it's an absolute must-own, either on Dreamcast (Japanese import), Gamecube, or Xbox 360 Live Arcade.
Genesis: Lots of great shmups! Some of the best are imports (the best of which are VERY expensive), but the US had some good ones. Thunder Force II, Thunder Force III, and Lightening Force should be at the top of the list, but don't ignore other titles like Bio-Hazard Battle, Trouble Shooters, MUSHA, Gaiares, Phelios, etc. The top import titles are Gleylancer, Eliminate Down ($$$$$! And rare, too.), and Twinkle Tale (technically not a shmup because it's not autoscrolling, but close enough).
Sub-Terrania isn't quite a shmup, but it is fantastic, and is probably the best Solar Jetman-style game ever.
If you've also got a Sega CD, get Robo Aleste, Silpheed, and Keio Flying Squadron. All three are great, perhaps fantastic. The 32X has Zaxxon's Motherbase 2000 and Kolibri, which are both okay and worth getting for 32X owners, given how few games the system has. 3d shooting games like Space Harrier and Shadow Squadron are the 32X's best, though.
PS2: It has a few released in the US... aside from stuff in the Midway, Namco, Tecmo, etc. collections, there's Gradius V, Gradius III & IV, Silpheed: The Lost Planet, Mobile Light Force 2 (Castle Shikigami 1), Castle of Shikigami 2, Raiden III, and a few more. Heavenly Guardian is shmupish too.
PS1 has some great ones here, too... R-Type Delta, Strikers 1945, Starfighter Sanvein, Mobile Light Force (Gunbird is still good, even this messed up), PhiloSoma (those longboxes are cool...

), In the Hunt, Einhander, Space Shot (super low budget game complete with subtitled Japanese voice acted CG cutscenes between levels... short and average, but entertaining), etc... some good stuff there, even without imports.
You mentioned you could get a Dreamcast fairly easily. There are plenty of shooters for both the Dreamcast and the Saturn. They are mainly imports though, and many are quite expensive. A lot of the games people are mentioning in this thread would fall into this category. The dreamcast has a list of very impressive and fun shooters but since you're just getting into the genre you might want to stay away from that sort of investment until later.
Dreamcast is great, but yeah, there were only four US-released shmups on it -- Mars Matrix, Gunbird 2, Giga Wing, and Giga Wing 2. All four are good, though, and Bangai-oh! is also of course fantastic. There are a lot more great ones among the imports... it's worth getting.
As for Saturn, it has only like three US-released shmups. In the Hunt, Layer Section, Darius Gaiden (fixed)... that's it. All of the system's other like sixty shmups are imports, and the better ones get expensive.
Oh, and Xexyz (along with other NES shmups like Gradius, Life Force, The Guardian Legend, Zanac, Gun Nac, Image Fight, etc) are great, but aside from the ones on VC, the OP said he doesn't have a NES. NES Gradius should be a must-buy on the VC though, along with Gradius III for SNES, Zanac, Axelay, all three R-Type games (particularly the first and third), at least one of the two Turbografx Star Soldier games, Gate of Thunder, etc.
flintstryker said:
and try overkill i fucking loved that game back in the days though i never got very far.
Great game! The shareware is like one level, but it's very hard to finish... awesome though. My favorite DOS shmup is Raptor, but Tyrian, Overkill, Major Stryker, Highway Hunter, Stargunner, and others are great games.