See
http://www.bbc.co.uk/proms/ . Bookings start on April 30. If you book early, you can get pretty much any one you want except the Last Night (and who would want to be seen dead there anyway

). As for dress code, you'll see loads of people in shorts and T-shirt, especially amongst the promenaders. This is a reflection of the standard of air conditioning, so if you're wearing more than that (which most, if not all, patrons do in the more expensive seats) you may want to leave your jacket in the hotel on a hot day.