How do I keep bubbles from coming up when I apply sheetrock mud?

Sarah you may be appying it too thick. Are you using the mud straight out of the box or are you mixing a little water in it? If your covering a tape joint spread just enough to cover the tape and let dry over night then sand it lightly and apply a second coat using a wider trowel. Next day do the same thing, let it dry overnight, sand and apply a third coat overlapping the second coat. This should prevent bubbles from forming.

Bubbles in joint compound occur when you hold the joint knife to flat. When finishing drywall, don't think of it as pressing the mud flat and smooth, think of it as cutting the top of the mud off smoothly. Hold the knife at more of an angle to the wall.

Also, get a "Whip" for a drill and mix it up really good... also, add some water. Check "You-Tube".. lots of demos on there