wich linux is the best to get wireless internet on my computer without much to do.
It all depends on the wireless card in the HP mini. Mine has an Antheros card. Took me 2 minutes to configure it, and I was on line. Some cards are much more difficult.
I would suggest you go with one of the major ones, like Ubuntu, even though I don't like the way they handle wireless cards. For most users, it is the easier choice. There is a very large community to help, if you have problems.