The jab is your most important punch; it's the fastest one to throw, most likely to land, and leaves you least vulnerable to a counter attack. When your jab is out there, make sure to keep your shoulder and rear hand high to protect both sides of your chin. Combine with a step forward to generate more power and use it to find your range. This will help set up your harder hitting rear hand.
You're going to get hit. However, if you learn to move your head properly you'll find it's much less common. Don't start wildly leaning from side to side at the waist when someone throws a punch, they'll follow your movement and time their strikes to hit you. Plus, you'll get tired fast. Instead, try small movements, slipping your head either inside or outside your opponent's punch. If done right they'll swing at air, leaving themselves open for punches of your own.
A properly thrown uppercut is an exceptionally effective punch. The punch can be risky as you'll need to drop your guard to throw it so make sure to set it up with one of your jabs. Designed to be thrown in close quarters, the uppercut's greatest strength is that it's hard to see coming: it comes from below and typically connects under the chin, all out of your opponent’s line of sight. The power comes from hip rotation, not from straightening your legs and lifting your hips up from the ground. If you can bait your opponent into ducking into it, they may well be seeing stars.