Good advice, especially making shallow passes. I never take off more than 1/8th inch or so at a time.
When inserting the bit, you don't want it to bottom out in the collet. My method is to drop a half inch GROMMET (not an O ring) and let the bit rest on that. Don't press it down. This will give you enough grip and absorb any vibration that might move the bit. Grommets are very precise devices. The difference between tight and loose a matter of a few thousandths of an inch. So tightening is very important, but excessive tightening can destroy the collet. Here's a picture of a collet's parts, you can see how the collet is made and how the nut drives it down into the spindle, tightening the fingers on the collet.
Good point on checking the fence. And is the top of the mounting plate perfectly level with the top of the table? Run the end of a ruler across the plate. If it grabs at any point, it's not really level.