How to drill plastic without the bit "grabbing" it?
Sandwich the workpiece between two pieces of wood, clamp it all together, clamp the whole assembly so that it can't go spinning off into oblivion, then drill through the whole thing.  You'll also want to use really light pressure when drilling through the plastic, but don't go so slow that you melt the stuff.

If you have some, you could use a step drill instead of a twist drill.



If you have a bunch of in between size bits, you can simply start small and slowly work your way up to the final size. However, you're still going to want to properly brace the material. Adding support behind the piece is also a good idea.

If it's within your budget, a drill bit designed to drill plastics might work. However, you're still going to have to secure the workpiece while drilling. I'd also recommend using a wood backer, to support the piece as you drill.