New to the Genre

[Deca]

MAME + DoDonPachi & Armed Police Batrider.

See you in a week!

This, have fun and welcome to the neighborhood

[BPzeBanshee]

Everyone's pretty much suggested all the games I'd go play except for ESPRade. It's hard for scoring gameplay if you're new but I like using it as a stress relief.

[MR_Soren]

I'd get DeathSmiles for 360, and if you like that, import Mushihimesama Futari. (Get the region-free version) Also get Raiden Fighters Aces because it is a great deal for some really good games.

SNES, I'd go with Space Megaforce (might be cheaper to get the import (Super Aleste) if you're on eBay. R-Type III: The Third Lightning is the other SNES game I'd suggest.

Wii, I'd get Gleylancer on the virtual console. Maybe "Ultimate Shooting Collection" which is cheap and contains 3 games developed by Milestone.

PS2, the Capcom and Taito collections are both good, IMO. 1942, 1943, 1943 Kai are all good, IMO. Metal Black is pretty awesome. I'd also suggest Castle Shikigami 2. Really fun game, really poor localization. Gradius V is also pretty fun.

MAME: Try GigaWing if you don't play it on your Dreamcast. Also try DoDonPachi and Battle Garegga.

[Deca]

I had a hard time getting into milestone games initially, but after playing a ton of other shooters and coming back to them I got hooked.

In light of this I would suggest grabbing Ultimate Shooting Collection while it's still easy to find (and cheap!). You might not care for it much at first, I know when I was first getting into the genre the last thing I wanted was a game focused more on chaining bombs than dodging bullets, but after getting way more than my share of dodging bullets I found the somewhat unique scoring system to be really interesting. Consider it a shooting investment :p

I should warn though, tate in USC is jacked, it rotates the controls with it and there's no way to tate normally :/ Still, much cheaper than picking up Karous and Radilgy on other consoles.