If you save after beating the game, will you have to start it all over..

If you save after beating the game, it is unplayable, so yes, you'd have to restart. However, you can save BEFORE you head to the top of the tower and fight the final boss, and just continue your game from there by leaving the Mars Light house and exploring or whatever you want to do. And if you do save, if you restart (In a different save slot, of course)you can actually replay it in a mode called Easy Mode which takes the save file of a completed game and takes the stats of your characters when you beat the game with and places them into your starting stats.

(Lets say you beat the game with the following stats:
Felix: Lvl. 54
Jenna: Lvl. 50
Sheba: Lvl. 51
Piers: Lvl. 53

When you restart the game, Felix will be Lvl. 54, Jenna will be Lvl. 50, and Sheba will be Lvl. 51. And, when Piers joins you, he will be at Lvl. 53. (The characters still level up)

This also applies to the original Golden Sun characters as well, assuming you didn't transfer when starting a new game (The stats might actually be the same as when you beat The Lost Age even if you do transfer, but I don't know because I never got the transfer to work for me).

This only applies to stats, so you will not get items or Djinn from when you beat the game, so you'll want to transfer from Golden Sun if you can if you want all the Djinn.)