Depends on what you like:
Bullet Hell? DoDonPachi.
Pattern/Gimmick Based Bullet Hell? Touhou 8: Imperishable Night.
Fast, Deadly, Bullets? Raiden DX.
Mixture of Fast, Deadly Bullets and Bullet Hell? Mushihime Sama
Slow Paced Shmup, A.K.A. the equivalent of Castlevania NES for the Shmup genre? Darius or R-Type games.
Two Player battle shmup, akin to Puyo-Puyo, but as a Shmup? Twinkle Star Sprites.
Puzzle/obstacle based Shmup? IKARUGA
Qusi-Shmup/Twinstick shooter? Blue Revolver.
High Quality Indie game with a bit of everything in it? Jamestown.
High Quality Indie game that's a greatest hits collection of Bullet Hells? Crimzon Clover: World Ignition.
My thoughts: If don't know what you want, go for Touhou 8: Imperishable Night, and get a V-Sync patch for it. It's an extremely beginner friendly Shmup that's more about avoiding patterns and gimmicks than raw bullet dodging. It has fun mechanics and well implemented difficulty levels, the latter of which is rare in most shmups. Getting the True Ending in that game is great practice for other shmups, and will give you an idea of what you like. If you like the bullet hell aspects, try Crimzon Clover next. Else, if you like the puzzle like aspects, go for Jamestown or IKARUGA next. If you like pretty little anime girls, go for Twinkle Star Sprites. (And visit a priest).
Things I would not start with: Ikaruga. It's great, but tough as nails and idiosyncratic. It's not representative of the genre.
DoDonpachi Resurrection. On steam and is good, yes, but has issues. There are better games.
Deathsmiles: Don't be a pervert.
Twinkle Star Sprites: It's more like Puyo-Puyo tetris than an actual shmup.
Blue Revolver: It's s Twin Stick Consoleshmup. (I think....) It's fun, but designwise it's 50% geometry wars, and 50% traditional shmup.