If you aren't ready to try jumping over obstacles, you can lighten the impact by unweighting the bike so you don't hit so hard. As the pothole approaches, stop pedaling, hold the crankarms horizontal, and rise slightly off the saddle. Support your weight evenly on hands and feet. Relax your knees and elbows. Keep your head up. Have you played basketball? This is the defensive stance. Be like a cat -- light on your feet and ready for action. Then use this bike-handling technique: Just before the front wheel reaches the pothole, crouch deeper and then rise up to unweight the bike. You don't have to hop if you're inexperienced or afraid. A well-timed upward pull with your hands and feet will lighten the wheels equally and lessen the impact.
Timing is everything. Practice to develop a feel for it. Don't start with real potholes. Use cracks or seams that aren't going to be a problem. Keep trying and soon you'll get the knack. This technique works best with small potholes and other imperfections in the pavement.
Now, what if the pothole is big? You'll probably get the front wheel over safely but bang the rear tire on the sharp lip. This can cause a pinch flat like you experienced, with the tube being popped between the tire and rim.
That's why jumping your bike over a pothole can be better in terms of wheel protection. Jumping means your wheels never touch a pothole's sharp edge. The trick is to assume the position just described but increase the upward lift so both wheels leave the road simultaneously. Your speed then carries you over the danger.
Jumping is rarely necessary, though. And unless it's done correctly it can damage equipment (or cause a fall) as surely as riding into an obstacle.
The landing is as important as the take-off for avoiding wheel damage. Flex your knees so they absorb the shock of wheels coming back to pavement. Practice this move on a grassy field by jumping over something soft like a rolled-up towel before you try it on the road.
Some roadies fly over potholes or railroad tracks just for the fun of being airborne. But jumping is a tool to use in critical circumstances, not for a cheap thrill.