Can't תיירת be thought of as a verb -- לתייר – "to tour"? So this would be "she tours in asia"?
Not quite, the correct verb is מתיירת. And it's not idiomatic to say היא מתיירת באסיה, although it would be understood.
And how it should be said: She tours in Asia?
Yes, in general, it can be tiyart(a) (you toured). However, it does not fit in this particular sentence, where תיירת is a noun.