The TG is my fave console of all time--way ahead of its time and underappreciated in the US. You can also play Japanese PC Engine games on it with an adapter cartridge (that simply switches 1 or 2 pairs of contacts on the HuCards, as I recall). I have a TON of TG/PCE/Duo games. And a TurboExpress handheld (plays the same HuCards). And a NIB shrinkwrapped Duo.
As for my faves, the shootemups on the TG are second to none. Great platformers too. From memory, here are many games I really like on the TG (many of which are already mentioned):
Shmups (they're all good on the TG!):
Blazing Lazers
Cyber Core
Dragon Spirit
Dead Moon
Galaga 90
Psychosis
R-Type
Soldier Blade
Super Star Soldier
Star Parodia (PCE)
Pinball:
Devil's Crush
Time Cruise II
Alien Crush (although the table is 2 screens high and flips instead of scrolls)
Platformers/rpgs/action:
Bonk, Bonk's Revenge, Bonk 3
Bloody Wolf
JJ&Jeff (funny! Japanese version, Kato & Ken, is a bit lewder)
Legendary Axe 1&2
Neutopia 1&2
Splatterhouse
Misc:
Chase HQ (excellent racer)
Victory Run (racer)
Moto Roader (racer)
Any Bomberman game
Klax (action puzzle)
Power Golf
Military Madness (*excellent* turn-based battle game, like Advance Wars on the GBA)
OutRun (PCE racing)