Victor Prasolov's Intuitive Topology is a translation of an eponymous Russian book (freely avaliable at http://www.mccme.ru/prasolov/ , but only in Russian). It is mostly about knots and homology, although at an elementary enough level not to presume knowledge of groups. I don't know of a good introduction to homotopical topology. Please don't pester western students with general topology; it is already way too popular over here.

I can't name a book right out of my head, but there should be some. At the moment I remember Etingof's introduction, but it is probably too compressed to be read by a highschooler without further instruction.

Ross Honsberger has many of these.