Congrats! I'm new to KLRs myself, but I also an loving mine. When you find yourself gravitating back to a particular bike it's a nice feeling when you finally get to take it home.
I can't speak on tires since by experience limited, but when I was asking around I kept hearing Kenda 270 for a good balanced tire that wears well, and Pirelli MT21 for a more aggressive tire that doesn't wear out horribly fast. But again, I've never ridden either. It can give you some names to research, though.
That said, when I first got my bike it has Shinko's street oriented tire on it. 705 it's called, maybe? I am coming to KLRs from inline fours running sticky 17" sport bike tires, and I must say, those Shinkos were amazing on pavement. Their dry traction is simply amazing... I drug a peg. In rain, they are good, but not great. If I was planning to ride pavement I'd definitely run those tires.