Fully agree with the previous posts.

For a 4+ month trip on the AT last year I took one pair of pants, one long sleeved and one short sleeved shirt (the latter a smartwool "town shirt"), one pair of underwear and a pair of shorts that work as both secondary underwear and pants to wear in town while washing the one pair. Two or three pairs of liner socks, one or two pairs of wool socks (two in colder weather).
That's it, apart from warmer outerwear which varied based on time of year.

Just make sure all of your clothing is synthetic (or smartwool if you're inclined), and quick drying stuff.
I didn't have a clean set of clothing for the trip home; I just put on my shorts and town shirt to wash my trail clothes and wore those.

How to wash besides at a laundromat: in the sink or bathtub in hostel or motel. Again, quick drying clothing, generally dry by morning (don't wash the thick wool socks unless you have access to a dryer ...). Note that some hostels and motels along the AT have a washer and dryer; it's always a good idea to ask at any rate.