Hi. A good way to see the Premier League is to head to Craven Cottage to see Fulham play a home game. It's a great old-fashioned ground with a fantastic atmosphere.
Tickets are under £40 and I'm pretty sure you won't have to book months in advance. Check out www.fulhamfc.com for details.
As for tickets to Chelsea and Arsenal, yeah, usually you do have to pay heaps and book ahead. However, recent Chelsea home games have not been selling out and the club has been taking out adverts to try to flog spare tickets.
Check www.chelseafc.com for details.