"I want to go."
Please note the difference between "Akarok menni" and "Menni akarok", both translated as "I want to go".
"Do you want to go to the zoo, or not?" "I want to go." – Akarsz menni az állatkertbe, vagy nem? – Akarok menni.
"Do you want to stay or go?" "I want to go." – Menni akarsz, vagy maradni? – Menni akarok.
You may also want to note the different use of punctuation marks, but that needs a complete book to explain all ;) Simply assume different manners if you see something strange.
First of all, "szeretnék menni" is more polite and shows a stronger dependency.
"Kívánok menni" is a kind of affection, it is more "I wish to go" and expresses a level of uncertainty.
"Akarok menni" is a decision, where I don't let the other too much possibility to hold me back, though they can stop me. In children/parent relation this form often triggers a reply "I will decide whether you go or not".
"Szeretnék menni" is the next turn for the child there. But it is not childish to say. It expresses that you are aware the situation and you're ready to accept the circumstances and change your plans. It is the right form if you tell your intention to a higher ranked person like your boss and usually triggers more emphatic consideration.
Kérek is asking for something. Literally, "I am asking for". So you can't use it when you want to go. That's akarok.
On the other hand, you can use akarok to ask for something, it is just very rude but not incorrect. :)
And kerek is "round".
When it is not about asking for something but, rather, wanting (to do) something, want has another alternative: would like.
"I want to go home" - "Haza akarok menni"
"I would like to go home" - "Haza szeretnék menni"
Indeed, pay attention to accents, they are modifying the meaning of the words! "Kérek" is "ask", "kerek" is "round", "kerék" is "wheel", "kérék" is ... well, it is an extinct past tense of "kérek". If the context is ambiguous, you can trick your English keyboard writing "ke'rek"—we will understand it thanks to the handelds' layouts :)
I don't understand why "Én szeretnék kérni menni." is wrong. Is it just weird grammar?
It doesn't make much sense. At best it means something like" I would like to go to ask".
Note that "I would like to go to ask" is an existing English structure with a completely different meaning. It is very far from the Hungarian example. But I think that many English speakers will fall this pit. "I would like to go to ask" is something "azt szeretném majd megkérdezni".
BEWARE MY FRIENDS, AVOID THIS MISTAKE ;)
"It's your own problem." will be the immediate answer. "Menni akarok!" (especially with exclamation mark) sounds harsh and selfish. "I will decide what do I do and you don't have any business with my acts, right? I want to go. NOW!" ("Majd én eldöntöm, mit csinálok, és neked semmi közöd hozzá, érted? Menni akarok! MOST!" – this Hungarian translation is very close in mood and expressive power but it is not a literal translation because that would not make too much sense.)