Saving properly in-game (as was done on the actual console) is done by going to the menu and choosing Save. OOT does let you continue exactly where you let off, but you'll always start in Kokiri Village when playing as young Link and in the Temple of Time when playing as adult Link. If you saved in a dungeon, you'll start back at the dungeon's entrance or before a boss room.
F5 and F7 save and load savestates respectively (or Ctrl+S and Ctrl+L if you want more control over them). Remember to choose an active save sta slot if you plan on using states, otherwise you'll overwrite/lioad the savestate each time you press F5/F7. I wouldn't ever rely solely on savestates, since they can be dodgy and get corrupted, but they have theiruses in a tough boss fight, for example.