You have two different verbs here so let's look at them separately.
The verb "want" has a very consistent pattern. It can be followed by a noun phrase or a verb phrase (with the verb in the infinitive). So you can make sentences like, "I want a coffee." "I want to visit Australia." When we use the infinitive verb phrase, either we want to do these things ourselves or we want someone else to do them. When we want someone else to do something, we put the object after "want" and before the infinitive. I've done it twice in the last two sentences.
So here are the possible patterns for the verb "want":
- I want [noun phrase]. [a new car, some bread, those shoes in the window, etc.]
- I want [infinitive verb phrase]. [to get rich, to go to NYC, to have a big family, etc.]
- I want [object] [infinitive verb phrase]. [you, him, her, my brother, the waiter, etc.] [to help me, to bring me a sandwich, etc.]
These are the only patterns possible with "want," it's not possible to say "I want going" or "he wants I be." All verbs have patterns of structures which follow them (you might notice "help" and "bring" in my last example: they have their own patterns too). I don't have a great resource for students to study these patterns, but I do have a rather technical resource for teachers: here is the section on this pattern of "want", here is the table of contents.
In the case of wish, please see this excellent guide to the different patterns (and their different meanings) here: Using 'Wish' in English - Articles - UsingEnglish.com