I wonder. What do I need to get higher level spells ? Do I just need to buy them, or do I need a certain level in a specific magic to be able to use them ?

You buy them and then watch them drain away your remaining mana like crazy, until you're high enough level to get perks that reduce the amount of mana they drain and when that happens you can use them.

They won't show up for purchase at all until your skill is high. Then they'll cost a lot of magicka until you buy the perk for them.