I like to use Swanstrom double-flush cutters, size 4. These are pricey but they cut right through the knot easily.
Before I bought those I used to use a razor knife or craft knife. These are quite cheap and can be bought at craft stores. Just be sure to change the blade often so it's sharp.
The best way to get a stuck knot out is to use a drill. I don't have one, however, and in the past have had to resort to attempts to pick it out gradually with a pin or needle. Very tedious and often not successful. I also have used a bead reamer but didn't like the way that widened the hole. If I were just starting out now with buying pearls, I would probably bite the bullet and get a small hand drill or multi-tool like a Dremel.