It seems odd to me. I can use my APs to move and then attack, but if I attack first then my turn ends and I'm not allowed to move.

Am I doing something wrong? Does this change later in the GF series?

That's how it's intended to work as attacking ends your turn. However in later games in the series the minimum ap to attack is removed allowing you to move much further while still attacking.

If your AP is high enough you can do multiple attacks or spells... Although maybe i found that by a glitch where i had 32AP...

You're right. Further into the game, I found that using Haste does grant enough AP to gain extra actions per turn.