I wanted to go a different skill path and so I went to a city looking for something like a class trainer but I couldn't find one. Now I wonder whether or not I can actually redistribute my skill points and and if yes: where and at what cost?
There are no class trainers and only the Divines will be able to help your poor soul - seriously.
You'll have to find a Shrine to Stendarr (or Shrine to Stuhn if you're in the Ebonheart Pact) to reset your skill points and apply them once again.
Keep in mind that this "service" requires a donation of some gold.
You're able to find such a shrine in each capital, based on your alliance. It's always located in the second mainland (post newbie island) zone:
- Aldmeri Dominion: Eldenroot in Grahtwood
- Daggerfall Covenant: Wayrest in Stormhaven
- Ebonheart Pact: Mournhold in Deshaan
If you'd like to redistribute your attribute points as well, you'll have to look for different shrines based on your alliance - they can be found in the same cities:
- Aldmeri Dominion: Shrine to Auriel
- Daggerfall Covenant: Shrine to Akatosh
- Ebonheart Pact: Shrine to Kyne