Everyone has their own system for drilling sea glass. The one thing I learned the hard way was DO NOT RUSH!!! You want to make sure that you are feeling calm and that you can take your time to get the sea glass drilled.
I have to admit that I am not the most patient person (I am trying to work on it!) and I shattered several pieces of sea glass when I was learning to drill, because I didn't go slow enough. Now, I make sure I am "in the zone" before I get set up to drill.
I use a dremel attached to a little work station. Once I have the piece of sea glass just covered with water, I hold the sea glass with a pair of my husbands long nosed pliers to keep it from moving.
I drill half way, flip the piece over and mark the drill spot with a marker. Then I complete drilling through the sea glass. It is always a great feeling to get through drilling several sea glass pieces without shattering any!