To say to a person "I want to make friends with you" does seem strange, but socially strange, not in terms of language.
It's not a good idea to write "wanna". Although native speakers pronounce "want to" as "wanna" very few write it.
I would find it more natural to say "I would like to be friends with him".
Dictionaries are fine for meaning, but they don't help as much with information about how native speakers use the language. That's why the forum here is so useful, especially as you will get answers from more than one country.