The key thing to understand about invisibility is that it's not just a simple vanishing act - there's quite a bit to it. You can grab these spells from illusion trainers or vendors at the mages guild once your illusion skill is high enough, with different versions lasting 30, 60 or 120 seconds depending on how skilled you are.

Here's what's interesting though - invisibility breaks the moment you do anything aggressive like casting spells or attacking, and it'll stop working if you try opening doors or talking to people. That said, you can still pick up and move stuff around without becoming visible, and your followers will keep tracking you even though they stay visible themselves. Watch out for magic traps too since they'll still trigger even when you're completely hidden.

If you're looking for something more permanent, there's actually a way to stack chameleon effects up to 120% using enchanted gear, which basically makes you permanently invisible. You'll need some grand soul gems and to complete certain shrine quests to pull this off, but it's pretty much game-breaking since enemies can't detect or fight you at all. Just keep in mind that some scripted events might ignore your invisibility, and if you're caught trespassing guards will still chase you until you leave the area completely.