great shmups :Gate4in1<- Must own. 4 great games, one usually low price, Lords or Thunder, Star Parodia, Terraforming, and Gradius II
Beat 'em ups are harder to come by- go for Riot Zone first (can be had for around $30 or less) and then save up for Downtown Nekketsu and Double Dragon II (Best version of DD2 IMHO- it's like a 16-bit CD tribute to the NES classic!)
for fighting games, its best you get an arcade card- with an arcade card you can get Fatal Fury 2 and special, Art of Fighting, World Heroes 2, and even a Kabuki (FEOE) Fighter. With that arcade card you then could find a cheap bootie of Sapphire from one of the guys around these forums (Sapphire is one of the BEST shooter experiences on the Turbo)
For RPG's I pretty much agree with everyone here- the US releases of Cosmic Fantasy 2, Exile (1 and 2), Ys, Dragon Slayer, Dungeon Master Theron's Quest, and Dungeon Explorer pretty much can't be beat. If you get through all of those go for imports of Ys IV, Xanadu, Gulliver Boy, and one of the best games ever made- Emerald Dragon!
and that is just for starters.....
You have a lifetime of gaming ahead of you, just be patient and the greatness will come knocking on your doorstep.... usually dressed in a UPS getup!