Please.

Radiant Silvergun, Progear, Ketsui, Thunder Force V for some of the sidescrolling elite.

Grand Cross Renovation, Cotton 2: Magical Night Dreams (english patch), Harmful Park for style points, despite being a little aways from watertight.

Gamera 2000 and Bulk Slash as 3D palette cleansers.

Kid Icarus Uprising if you have 3DS around.

Harmful Park if you want one that’s just about having fun.

Parodius if you want something with the same vibe as Harmful Park but harder.

Radiant Silvergun’s Xbox/Switch ports if you want a tough shmup that you can be dragged through with a system that rewards you with training wheels for trying sequences again and again until you master them.

Dariusburst: Chronicle Saviour if you want a shmup with console-friendly “campaign” mode that weans you onto its game systems and has this amazing otherworldy vibe + music.

Eschatos if you want something with a unique vibe that’s on the harder end of the spectrum but simple to learn the mechanics of.

Sin & Punishment: Star Successor if you want a 3D shmup.

Disc Room if you want a shmup that isn’t a shmup.

Gate of Thunder for a banger I've played recently.
R-Type Delta for some very very good atmospheric horror bits.
Gradius Gaiden if you want a very good Gradius.
Einhander, because I just think it's really cool.
Thunder Force III if you want something easier, but with cool music.

Sorry that the bulk of my suggestions are PS1, I just happen to really like those games. Also have a heavy preference for being horizontal.

Darius series is all cool. G, Gaiden, the MD versions of 1 and 2, and Twin are all good.

Sexy Parodius
Space Invaders '95: Attack of the Lunar Loonies
All Gradius and Salamander, especially Gradius V

Really appreciate these so far, folks.

Might I recommend Touhou Shinreibyou ~ Ten Desires, the 13th game int he Touhou project, and arguably most beginner friendly?

Zanac x Zanac is a bit long, but quite forgiving and has an awesome soundtrack.

Enemy Mind is worthy a try, it's a reverse shmup where your mind jumps from enemy ship to enemy ship and a huge focus on the narrative.

Border Down has some unique death/difficulty mechanics and a soundtrack that's peak dreamcast-core. I'm still in the process of 1cc'ing it, but it's actually not that hard, I'm just taking my time with it.

Metal Black is a broken and unforgiven mess but COOL AS HELL.

Gradius: The Interestellar Assault for GB is short, accessible and very underrated.

If you can withstand the choppy scrolling, the MSX versions of Gradius are quite creative and actually technically impressive.

Magical Chase
Also for Parodius I'd say start with Gokujou instead

Graceful Explosion Machine is a bit Defender and a bit Geometry Wars and a bit addictive

Gynoug on the mega drive/switch is cool, nice gothic horror movie vibes.
Bangai-O Spirits is a good way to see the insane amount of sprites the DS could handle.
Geometry Wars: Retro Evolved 2 is twin stick shooter perfection

MUSHIHIMESAMA

Great shmups that i think you should give it a try:

Ether Vapor Remaster (steam)
Danmaku unlimited 2 (steam/ Mobile)
Thunder Force 4 (Megadrive)
Sin & Punishment (n64)
Judgement Silversword Resurrection (steam)
Battle Mania 2 (megadrive)
Lords of thunder
Gradius 3 (snes)
Space invaders extreme