Micha,
After looking at your site and running a few quick tests, I still don't think your site has been penalized. You have a few technical and on-page issues to deal with, but I don't think fixing these alone will catapult you to the top of the SERPs.
I checked the SERPs for "climb kilimanjaro," and I think your competition is your biggest hurdle. In addition to being around a lot longer, your competitor's websites have far more targeted, editorial backlinks, and consequently much higher authority, than your site.
Unfortunately, there's no quick and easy fix for this problem. You'll have to work toward earning good backlinks. Check out the resources below.