I have done the tutorial in the game. Do I have to level up to use more than 3 memorized spells at a time?

I'm not sure if you are playing NWN 1 or 2, but both tutorials leave you at level 3. I can't remember exactly how many spells you should have especially since I don't know what class of spell caster you are using, but you need to go to your spell book to select new spells when you level up.

You can get a fex extra spell slots for high ability scores that are tied to the class: intelligence for wizard, wisdom for cleric, etc. You can also get an extra spell per level if you chose to specialize in a school for a wizard - but only for a spell of that school. There is also a few items you can find/purchase in the game that will give you some extra spell slots for certain levels.

Spellcasters are a little hard to start in the game because of the lack of spells, but become quite powerful after getting a few levels.