Not sure about tent pitches, but they aren't too hard to come by. Anywhere with a beach should be good.
The obvious hill is Clisham, but including the ridge over to Mullach Bho Deas makes it a much better walk. Any of the other 2000ft summits are worth the effort if clear as well.
You can pretty much go up anything over 100m in height and get great views, so you are spoilt for choice. If the higher hills are in the cloud (or even if they aren't), Ceapabhal (near Northton) is an absolute must. It sits just off the main body of the island and has some of the best views of the west coast beaches. Beinn Dubh just south of Tarbert is also highly recommended (have to admit I haven't been up it yet), as is Todun on North Harris.
You should also take a trip down to the road-end at Hushinish on North Harris for the beaches and coast there. Just noticed you have no car, but the bus service isn't too bad for such a sparsely populated area. Hitching is dead easy on the main roads, but on quieter roads like the one out to Hushinish, you can go for the best part of an hour without seeing a car sometimes. Bus timetables here if you haven't found them yet:
https://www.cne-siar.gov.uk/popular-tasks/bus-services/I'm jealous - wish I had a week on Harris. Have fun!