There's a lot of theories about correct ways to pedal and different ways to do it, and different equipment setups that can either help you do it right or help you learn to do it right. But unless you're worried about small differences (like for competitive riding), it can be kept relatively simple. I'm sure we have some users who can talk more about specific pedaling techniques.
- Pedal in an easier gear (but faster).
- Try to use more of the pedal stroke.
- Don't fight your own pedal power.
- Don't stand. Stay on the saddle.
- Generally try to keep everything smooth and even with no bouncing.
The easier gear thing should be fairly obvious. You can get the same power by pedaling faster in a lower gear. This should reduce the strain on the bottom bracket area. Don't power through a tough gear when starting up, downshift while slowing down.
There's a tendency for some people to "mash" the pedal from a bit in front of the top of the stroke down to about the bottom. Often with a substantial side-to-side weight shift. That will always be the most powerful part of your pedal stroke, but try to also use the top of the stroke and the bottom. That kind of mashing seems likely to be harder on the frame. Try to push the pedal over the top and scrape it across the bottom.
As far as fighting your own pedal power: on the back side of the stroke, try to lift your foot a bit. Don't lift so much that your foot actually comes off of a platform pedal (this is an advantage of clips or clipless systems). Most people leave a fair amount of weight on the back pedal and they're pushing against it with the front pedal. If you provide some lift to the rear foot you'll go faster and produce less total downward force on the bottom bracket. Especially if you're clipped in, it might feel like you're pulling the rear pedal up.
Standing puts your weight all on the bottom bracket and handlebars. The bike is designed to have a good portion of your weight on the saddle. Occasionally standing shouldn't be a problem, but doing it too often is a common bad habit and hard on the bike.