What's the lore behind death in the Elder Scrolls universe? Do they go somewhere as spirits? Get recycled?

Spirits can get recycled and put into new bodies without any memories in a thing known as the dreamsleave.

They can also go to a Daedric Realm provided they have served that prince sufficiently, or by virtue of what they are e.g. Apparently wearwolves are drawn to Hircines Hunting grounds after death.

Nords who died valiantly in battle can go to Sovenguard, a kind of Nordic valhalla.

Also, provided your angry enough with your death or you have unfinished business you can hang round as a ghost.

Also a suitably skilled necromancer can bind your soul to them and you'll live as a subjugated ghost. And suitably skilled necromancers can apparently rescue your soul from wherever it is with the intent of putting it back into a body (see Potema in Skyrim).