October half term holiday cottage by public transport

[Temeraire]

I want to get away for a few days over half term but can’t drive. I fancy a small interesting town near country or seaside where we can chill, go for walks, eat out when we don’t fancy cooking, do lots of reading/revision/homework and basically escape London - within about 2 hours by train.

Rye, Whitstable, Sheringham, Lyme Regis are the sort of places I’m thinking of are, but there are probably some nice cheaper non-coastal places I haven’t thought of.

Irritatingly none of the sites let you filter by “accessible by public transport”, so can you suggest places I should be looking?

I’d go to Brockenhurst in the New Forest. It’s very pretty and there’s a SenSpa at Carey’s Manor. Grab a bicycle from Cyclexperience, where you can set off on one of their recommended routes. These include rides to Beaulieu and Buckler's Hard and a route following a former railway to Burley.
Setley Ridge Vineyard and Farm Shop is a brilliant place to stock up on local produce for picnics and to find ideas for evening meals. Brockenhurst is dotted with little retro clothing and antique shops that are worth a visit.
Just outside Brockenhurst is one of the best eateries in the New Forest, and it’s called The Pig. You will need to book a table but it’s well worth planning in advance. If you like walking, head to Balmer Lawn. Many of the local trails originate here and from around the Standing Hat Woods. A pleasant place to find New Forest ponies and enjoy the joys of the local countryside. Follow the circular route which is very short and suitable for people of all ages.

We don't have a car. I'd recommend Bournemouth. Also Ryde on Isle of wight. Bit further, but worth it.

Thanks both. Brockenhurst sounds like a great shout and has given me a steer to Beaulieu and Lymington as alternatives or perhaps Winchester - I’ve got a bit of a blind spot on Hampshire. I instinctively feel that Bournemouth is too large but will give it a look as well.

Anyone else got any great ideas?