If you are going to try the delights of East Anglia.....what I did a couple of years ago (in February...):
I live almost in Gatwick Airport. Train from airport to London Bridge. Ride North to Liverpool St station. Train to Ipswich. From Ipswich, through Woodbridge to Dunwich. Over little bridge to Southwold (Adnams Brewery), then Lowestoft, Gt Yarmouth, round coast through Caister and inland past Hinckling, through Broads to Stalham, up to Cromer and back down to Norwich. Train back to Liverpool St, ride to London Bridge, train to Gatwick.