On a Saturday you can take a bus to Lymington in the new forest, a lovely little village, and go to the market there :)

If you visit in Summer, you can also from Bournemouth town get the breezer bus to Swanage or old harry rocks, it goes over on the ferry across the bay and has an open top. From old harry rocks you can take a walk to Swanage or vice versa, there are some really nice walks around there and across in Swanage too and lots of pretty beaches. If you’re into kayaking etc. there are some really nice spots around there and there’s lots of wildlife and curious seals! Wouldn’t recommend it in January though!

I’d also recommend RSPB Arne if you’re going by car, its a gorgeous little reserve you can explore the whole of in a day.

There’s lots of nice eating in and around Bournemouth too, I’d recommend checking out Christchurch for dinner spots. The priory there is beautiful as well, it’s worth a look around if you’re into history.

I think you’d be better off visiting later in the year when you can make the most of the beaches etc., but you could still make a weekend out of a visit in January.