... to use the move stat, does it still cost you a gem to get the move bonus like all the other stats?
Your question is somewhat unclear (to me, at least), but sounds like something a quick reading of page 11 of the Heroes' Book would answer. If you mean OL minions, see page 4 of the OL book (after reading page 11 of the Heroes' Book).
For Heroes: The first time you declare a move action each turn you do not need to place a gem in the movement box to gain your 'Free' movement points. Any subsequent move actions that same turn cost one gem per space moved. Note 1: It has been clarified that the rulebook is a little unclear of when 'Free' movement can be used. It can be used at any point during your turn as long as it's your first move action that turn. Note 2: If you do not use all your 'Free' movement in a single move action the remainder is lost.
For the Overlord: When you activate a tile on the river all units for that tile can move the tiles stated movement points without the overlord placing gems in the movement box. If the Overlord would like to move these units further they may spend one gem per unit, per space. Note 1: The Overlord must move all units for that tile before declaring attacks. Note 2: A unit cannot gain extra movement points past it's 'Free' movement. Example a Pirate has 3 'Free' movement points, so can only have a total of 3 extra movement points payed with gems. Note 3: The Overlord can spend gems for payed movement before or after a unit attacks (this is the only way to a unit after it attacks).