I am a new member here too, welcome.
I would urge you to purchase Ninja Spirit. It is a game that is on the same level of excellence as Ninja Gaiden (since you said you liked the NES) and Sega's Shinobi series.
I think you should also get Keith Courage, really just to have it as an introduction to the Turbo. Go in with expectations low and you should have a great time. It is a lot better than it's reputation, it is the most inexpensive game for the Turbografx, and it is what everyone got as a pack-in circa 1989 that introduced us to the system.
R-Type is a great shooter on the TG-16 along with Raiden for the more common titles. Air Zonk, Soldier Blade, and some of the other releases can start to really break the bank (but are worth it). Blazing Lazers is a great choice for your first shooter on the system. It was made by Compile who also made Zannac on the NES.
I really like New Adventure Island per platformers, which is an update from the same named NES series. Dragon's Curse is well worth playing if you do not own the Sega Master System version.
I get a lot of wear out of Military Madness as a strategy game. The first CD game you should find is the 3-in-1 CD with Bonk 1, Bonk 2, Gate of Thunder, and Bomberman if you have the 3.0 system card. If not...I am not sure if any of that will run on a 2.0 system card. Senior members can probably help with that question, but it would present a way to save a lot of money and get some excellent games. A mandatory purchase for the CD unit is the Ys (I & II) disc, which will run on any system card.
EDIT: Dang typos, they are always tripping me up.