Crimzon Clover, any version's good but World EXplosion is the most recent. It's a fairly difficult and chaotic bullet hell, but the novice mode should be reasonably approachable as long as you're willing to learn, and the design is superb.
Similarly, the whole CAVE backlog. Not all of them have novice modes or the like, and there's quite a few games not really available outside of MAME. The original DoDonPachi is/was considered the best starter bullet hell for a long, long time and still holds up pretty well, but is more difficult than a lot of modern games on their respective novice modes.
On the indie side of things: Star of Providence (formerly Monolith) is an indie roguelite bullet hell twin-stick-ish shmup with a pretty good amount of depth. ZeroRanger is a much more story-based game that I really enjoyed.