Well I found myself getting to the last boss in Bangaio on the Dreamcast, even though I was never that much of 2-d shooter player. It's not quite your traditional shooter, but fun and having a life bar and the ability to counter swarms of bullets with one of your own helps keep the difficulty down.
I also like Sexy Parodius if you can play imports. It's fairly easy to pick up and play and not too difficult for a first timer. Given how funny, wacky and of course, sexy the game is, you'll probably want to keep playing even if you do die.
If you had a modded Saturn I'd say Radiant Silvergun. That sounds like an insane recommendation given the game's notorious difficulty, but you get more continues and stronger weapons the more and more you play, and each time you start the game again you have the option of carrying over your stronger weapons and continues.
It's brilliant, and really encourages you to keep trying as the game is essentially helping you and rewarding you for your persistance. Plus, the game is actually pretty easy to play control-wise. Nothing too complex
Newb shmups?
I definately concur with Gigawing. You can find it on mame or the DC. Both versions are the same (when emulated properly).
Very easy to get into. The bullet reflecting is a nice little gimmick that helps getting out of tight spots. Especially when you can't be bothered with bullet dodging.
Also try 19XX or the earlier 1945 games. Very simple. Challenging towards the end but never completely insane. It's more about dodging all the craft, blow up everything in sight rather than dodging thousands of on screen bullets.
Very easy to get into. The bullet reflecting is a nice little gimmick that helps getting out of tight spots. Especially when you can't be bothered with bullet dodging.
Also try 19XX or the earlier 1945 games. Very simple. Challenging towards the end but never completely insane. It's more about dodging all the craft, blow up everything in sight rather than dodging thousands of on screen bullets.