In the first game, when you get to the Yellow Devil boss fight, you can use Elecman's ability, and then mash select to pause and unpause the game which makes the move hit a million times, making a super tough fight real easy.
In Megaman 2, almost every boss is weak against Metalman's ability, so get him first if you want to have an easier time.
In Megaman X, do Chill Penguin's stage first to obtain the dash boots as early as you can, it'll make the entire game tons more fun.
EDIT: Oh! The biggest tip I can give to anyone trying out Megaman games for the first time is, enjoy yourself. They can be pretty tough, but they're also really tightly designed for the mostpart, so if you pay attention to the layouts of the levels you can usually come out on top once you learn patterns and whatnot. And there's no need to play them in release order if you don't want. Play the ones you think look the coolest and see what sticks.
Man, I wish I could play Megaman for the first time again.