There are many fun things to do and nice things to eat in St. Leonards and Hastings. Let's start with St. Leonards. If you come out of the building and, facing the sea, turn right you can walk along the coastal path that is a nature reserve. There is an ancient underwater forest, a shipwreck, and a lesser spotted Retail Park that includes a & Spencer! If you're feeling super fit you can walk to Bexhill and see the lovely pavilion; and if you're feeling bionic, why not walk all the way to Beachy Head? (Make sure you have a mode of transport other than your legs for the return journey.)
If you turn left out of the building and head east towards Hastings...
...Grosvenor Gardens has a bowling green and a putting area.
St. Leonards was founded in 1828 on a deserted stretch of coastline a few miles west of Hastings, built by James Burton - the most significant builder of Georgian London, including Regents Park which he modelled the beautiful St. Leonards Gardens on. In Burton Terrace is the house with the blue plaque where Queen Victoria stayed, before the Royal Victoria Hotel (where she allegedly stayed). Behind this you can get access to the gardens.
If you carry on you'll reach Marina Court, built in 1936 to look like an ocean liner, and was the tallest block of flats in the country. Needless to say, this was a dream for WWII German bombers to get their bearings, and word has it that a replica was created out at sea as a decoy. Now, you can find the best independent burger joint in England below it - Half Man! Half Burger!. And a selection of lovely vintage shops and cafes - it's constantly evolving.
Other notable restaurants along the seafront is the family run Italian La Bella Vista - very authentic, delicious food with lovely friendly service (and of course great views!).
Turning left up London Road takes you into the heart of St Leonards - turn left again into Norman Road for the cool independent shops in the area known as Mercatoria. Here you will find the incredible Kino Teatr - cinema, restaurant, art gallery, music venue - and also with a great history (ask the friendly staff for the story of how it became the Kino Teatr!)
Going back up London Road (via the excellent St Leonard Pub), then right down Kings Road you'll find a lot of great vintage/charity/thrift shops, a great Fish n Chip show and some nice bars and cafes. This will take you to Warrior Square train station where you can go one stop either into Hastings or one stop back to West St. Leonards where Wilton House is.