If you're just starting with shooters, it's probably a good idea to explore them according to developer. See what style klicks with you and which one doesn't. Maybe check them out in MAME first, etc.
Some major developers:
Gradius series and lots of games similar to Gradius like the parody Parodius series.
- Gradius V (PS2)
- Salamander Deluxe Pack (PSX, SAT)
Lots of hard as bones oldschool tactical shooters with high addiction, as well as frustration factor. Kind of love-em or hate-em, although even the people who love them seem to be in kind of a love-hate relationship with them.
- R-Types (R-Type I&II), PSX
- R-Type Delta, PSX
- Image Fight & X-Multiply, SAT, PSX
Masters of the straightforward, no-frills blast & bomb concept. Batsugun was their last game and really set the stage for later bullet hell games. A bunch of Toaplan staff later went to Cave.
- Batsugun, SAT
- Toaplan Shooting Battle (Tiger Heli & Kyuukyoku Tiger), PSX
Complex score shooters that adjust the difficulty quite radically according to your performance. Some people swear by them some people hate them. Either way, they're always really nice to look at. Head programmer Shinobu Yagawa now works with Cave, where he made the Raizing-inspired Ibara.
- Battle Garegga, SAT
- Soukyuugurentai, SAT
- Ibara (Cave), PS2
Well, you already know about them. Their most enjoable ones on console would be, IMO:
- Dodonpachi, PSX
- Espgaluda, PS2
Super fast, simplistic games that require you to memorize bullet patterns. Psikyo were really prolific and a ton of games abound on all platforms. Some claim they essentially just made the same game over and over again, only with different graphics.
- Strikers 1945 II, PSX
- Sengoku Blade, SAT
- Zero Gunner 2, DC
Manic score shooters, were dodging is often impossible, instead one needs to use a well-timed bullet reflector. Probably the best current arcade shooter developers next to Cave.
- Mars Matrix, DC
- Giga Wing, DC
Taito & G.Rev
Lots of really atmospheric and dark shooters. Layer Section (Rayforce) really broke a lot of ground in terms of score-play, and IMO it's the finest Shooter on Saturn (and luckily also the cheapest). When Taito stopped developing arcade shooters some staff split off and formed G.Rev to continue the legacy.
- Layer Section, SAT
- G.Darius, PSX
- Border Down (GRev), DC
- Under Defeat (GRev), DC
Fast and super-smooth oldschool action. Everybody knows Thunderforce right?
- Hyper Duel, SAT
- Thunderforce V, SAT, PSX
Well you know about them and their 2 shooters.
Makers of the Raiden series - and only the Raiden series. Pretty playable games, but I don't really understand what the hype is about. Ton of better games around.
- Raiden Project (Raiden I&II), PSX
- Raiden DX (basically Raiden II with am awesome scoring system), PSX
- Shienryu (developed by Warashi, supposedly ex-Seibu staff? Anyway, this is a lot mor awesome than any of the Raidens, IMO), SAT, PS2
- Einhänder, PSX
Probably the best shooter on Playstation. Developed by Square of all people. Really atmospheric, technical and memory based, but also has some cool scoring tricks.
- Harmful Park, PSX
Awesome Cute 'Em Up with superb 2D graphics and gameplay.
- Twinkle Star Sprites, SAT, DC, PS2
Ported from Neo Geo. Has the revolutionary concept of a Vs. Shooter. Tons of fun playing against a friend. A Must-Have!
- Psyvariar series, DC, PS2, XB
Amazing psychedelic bullet patterns and score-play involving "scratching" bullets as closely as possible.
...man, that was long. Well, I hope this helps you in exploring the genre.