Suspension bikes don't like it when you mash on the pedals and will tend to bob as you ride. You may need to smooth out your pedal stroke, maybe even pedal a bit faster as slower cadences can cause the bike to bob.
On rigid frame I can stand and lean forward and sprint hard on the pedals, concentrating on putting all my power down through the pedals. On a full sus, unless you have lock out, this isn't going to work and hard sprinting out of the saddle is going to have the bike bouncing all over the place. When I sprint on a full sus, I don't lean forward, in fact I pull back on the bars counter acting my down push on the pedals, almost unweighting the front end.
In corners the bike will give and compress into the bend. You can learn to maximise this and really push into the corners where on the exit the suspension will uncompress and give you a small 'pump' forward.
Let the suspension work, don't over pressurize the shocks to make the bike stiffer. I usually aim for about 25% sag in the suspension when I'm just sitting stationary, yet still allow for enough movement in the shock to bottom out on really big hits.
Get spds and chuck the flats. Nothing to do with suspension, just a personal preference of mine.