What do you like doing?
Woodbridge, in Suffolk, is an old-fashioned town, with a stunning section of the River Deben alongside its tide mill, and easily reached by train (a few direct, but otherwise a change at Ipswich). From there, you can reach must of the Suffolk Coast rather easily (Aldeburgh, Orford, Thorpeness, Southwold) - a place that has recently become one of my favourite spots in the UK.
If you fancy the South Coast, one of my favourite spots down there is Bosham, where Canute managed to prove his point. You'll have to change in Havant or Chichester and walk a bit, but it's a really lovely spot, with some nice boating in the Chichester Harbour (a good cycling spot, too). Chichester itself is nearby, a nice little town with a lovely cathedral and some Roman ruins.
York is an all-round great city, I love it there. It's quite far from London (200 miles) but the train is fast enough (under 2 hours) to make a day-trip doable. You do need to get the cheap tickets, though, as close to 12 weeks in advance of the trip as possible, or it'll be rather costly.
Stratford-upon-Avon is somewhere I've always fancied going - and it's quite easy now from London Marylebone by train, with tickets from £5 in advance.