There are two standard ways to move in doubles. At the advanced levels, if you have two righties and they train regularly for doubles, they usually learn circling footwork, where after each shot the player steps backwards and circles clockwise to the left so they can approach the table from their backhand side and go into a neutral or forehand-favoring ready position. (A righty's normal ready stance would be toward the left, with his playing elbow near the center of the table, or more to the left to favor the forehand, or more to the right to favor the backhand.) Two lefties would do the same, except they move counter-clockwise to the right. However, learning to do circling footwork takes a lot of practice, usually under the supervision of a coach. You also need two mobile players.
There are four movements made with this circling footwork. Assuming two righties, they are:
For most players (and for lefty-righty combos, which is an advantage in doubles), it's easier to use in-out footwork. Basically this means that after each shot, a player steps backwards to allow his partner to move in, and the two take turns, moving in and out. (You don't want to move too much to the side or 1) you'll be out of position on the next shot; 2) if the opponents hit an angled shot toward you you'll likely block your partner, and 3) if you don't block your partner against an angled shot, he'll likely block you from getting back after that shot.)
The disadvantage of in-out footwork is that if there are two righties or two lefties, one of them will usually approach the table from the forehand side, and so won't be in a natural ready position. This is fine if one of the players is stronger on the backhand than the forehand. So you should normally position the stronger forehand player on the left, the stronger backhand player on the right. (Reverse for two lefties.) If you have a left and a righty, then in-out footwork is natural, with the lefty trying to stay on the right side, the righty on the left side.
There are four movements made with this in-out footwork.
You can also do a hybrid of these two styles of footwork, where you generally use in-out movements, but when you see the opportunity, you circle around to the left (for two righties) behind your partner so you can approach the table from the backhand side and so get into a neutral or forehand-favoring ready stance for the next shot. With lefty-righty combos (or if you have one player much stronger on the forehand, one much stronger on the backhand), you would favor in-out footwork, but perhaps switch to circling footwork when the two players are stuck on the wrong side so they can get back to their better positioning.