Better tires are always "better". Street tires will spin easily off road, especially if it gets wet. I have a set of 5 Goodyear Adventures off of our Sport with only 1650 miles on them. $200 for all 5. The bad news is I'm 20 miles west of Gettysburg, PA.
Yes get some all terrain tires.. goodyear duratracs, bfg ko2's etc even in stock size will make a big difference. Load range "c" have weaker side walls than "e" but will ride less stiff.