My favourite #shmup might be Dariusburst: Chronicle Saviours at this point. So much to do, great presentation, fun scoring based around effective beam use.
I've always liked the genre but it was Ikaruga on Dreamcast that really clicked with me, probably because its mechanics are a lot more transparent than many shmups.
Danmaku Unlimited 3 and Crimzon Clover: World Ignition are great entry points. They're solid introductions to bullet hell without being utterly overwhelming.
Batsugun is my answer to all your questions, at least for vertical scrollers. For horizontal scrollers, Death Smiles is my fave.
Those are the two games that got me into the genre. Now I'm a bit obsessed, even though I still pretty much suck at them.
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