If hour building in USA don't limit yourself to one state - fly across America, up to Canada and back again. I did that - buy a road map and decide where you fancy going - we saw some great places; set out early in the morning, stop for fuel then fly on and spend the afternoon somewhere interesting sightseeing, then fly again for a couple of hours in the evening. Get to know 1-800-wx-brief people really well, particularly the met man!
Tips for saving money:
1) put advertisements up in flight schools in the UK asking if anyone wants to come with you - many newish PPLers want the experience but don't want to go alone - share the costs, just make sure you log all the time. Can make the trip more fun.
2) get a copy of the AOPA guide (I think its AOPA, looks like a telephone directory). It lists all airports in the USA, with useful info such as whether they provide free courtesy car, free "pilot rest area" (which is basically just like a motel room, for free), etc.
Many do. Some are fantastic (try Austin, Texas).
3) try and find out where the cheapest fuel is to be had. Many pilots will know if they've been somewhere and it was cheap.
4) do as many landings, night flying, instrument approaches, approaches to big airports as you can as its relevant to airline flying and its free
5) when staying in cities stay in the affiliated youth hostels, dirt cheap.