Since this is called 'shmup chat' and the word chat sounds so open and inviting, I will ask: What would you recommend to the total shmup n00b? System probably won't matter, but let's keep it affordable.
For some that won't cost you a fortune, and are truly timeless: (and accessible to someone new to the genre - ie you won't die every 3 seconds like a bullet hell game)
Sega Genesis - thunder force series
Snes - axelay, super r-type,
Ps1 - r-types
Sega master system - astro warrior
Xbox 360 - radiant silvergun (only because it's bloody expensive to get the saturn version)
Amiga - hybris, battle squadron, swiv
In honesty I could go on and on, but those are some of my favourites.