If you just want woodland walks where you drive to and park up then here's a few I can recommend:
Brede High Woods (between Cripps Corner and Broad Oak)
Guestling Wood (Watermill Lane, Pett village)
St.Helen's Woods ( St.Helen's Park Road, Hastings)
Battle Great Wood (Marley lane, Battle)
Camber castle is a nice walk too, starting from Harbour Road, round to the castle and back along the River.
For a longer proper Hastings walk, check the tide times and do a Fairlight beach loop.
Start at Hastings station and walk through the Old Town until you reach The Dolphin pub. Nice...
As others have said, that Queens Head one is nice.
By train - From Winchelsea Station head up to Winchelsea via Ferry Hill, along Monk's Walk and across the fields up to Hogs Hill. Over Hogs Hill by the windmill and down into Icklesham and onto the Queens Head. After your pint, or two, head...