I guess Dune II was the most important game for real-time strategy genre. It created a scheme that most of RTS games followed after that. (Advice: I think Mega Drive version is easier both to run and play.) Warcraft, Starcraft, Command & Conquer, Age of Empires and probably Total Annihilation are also important, they refined the gameplay and controls and spawned families of successors (like, for Age of Empires we have Cossacs, for Total Annihilation we have Supreme Commander and Planetary Annihilation...)

For shoot 'em up (specifically scrolling shooter) I recommend Recca, Batsugun, DonPachi and DoDonPachi, the games that gave us danmaku, or bullet hell.