From Normal to Legendary is quite the jump actually - hmmm.
Well, remember that the enemies are more accurate to the point that they're auto-aiming at your head, so stay in cover a whole lot more than you're used to, always go for headshots, try to stick to precision weapons for those headshots, Needle Rifle is the anti-unshielded enemy gun especially to Brutes (three shots anywhere is a needle explosion kill), Plasma Pistol Overcharge x 2 + Headshot is the anti-Elite solution, and don't rely on your friendly AI anymore, it's all you now.
That's the basics right there, you'll get the feel as you go along with those starters...