"I do not want to go."
Translation:Nem akarok menni.
"Kérni" is surely not given as "I want", it's an infinitive form, pretty much like "to want". (Or are you referring to "hints" by any chance? In that case, I'd say don't rely on them, like, at all. Contributors aren't free to change "hints" and they are very clearly optimized for the reverse tree.)
"Kérek" (when used with something not a person in the accusative) is like "I ask for". It cannot be used with an infinitive at all but it's the way to go when you really just want someone to give you something.
"Akarok" is literally "I want". It works the same way but it's highly impolite to use when you could use "kérek". You will come off as rude if you say "akarok egy süteményt" in a pastry shop.
"Kívánok" is literally "I wish", including making wishes, expressing desire and wishing someone a happy birthday or similar. Theoretically, nothing rules out that it could be used with an infinitive but it sounds rare and fairly pretentious. Just like "I wish to go" or "I desire to go" would sound awkward most of the time. Probably a bit worse.
I would be curious how you collected these words, then I could answer how you are supposed to know the difference (or are you supposed to know it at all) if you are still interested.
As jzsuzsi said, all are in the same lesson. And when I saw "szeretnék kerni," knowing "kerek" meant want or would like (same meaning in English, the latter more polite), I thought "kerni" was a differently inflected form meaning "I would like." Duolingo doesn't teach us grammar, so I've never been shown an infinitive. Thank you for the distinction among akarok, kérek, and kivánok. I wish Duolingo would tell us this instead of translating all as "I want" without explanation.