My daughter is at University in Bournemouth so over the past 4 years or so we have spent quite a bit of time down there for the occasional break here and there.
Bournemouth is actually quite nice if you want a decent beach, but public parking is horrendously expensive. Parking for just a couple of hours just to go and have something to eat can cost close to £10
The Seaworld type place is good fun
There is however a very good choice of accomodation at reasonable cost and we now find it easier to leave the car at the Hotel and get a Taxi.
From a what to see point of view, other than above, Poole is probably a better bet, even maybe Swanage for the steam railway, Swanage castle, Bovington Tank museum, and there is a nice fish and chp shop on the quay side in Swanage. Weymouth might be a better option.
I would suggest Brighton, but I have to spend quite a bit of time down there with work and it is another horrendously expensive place to visit.
As a completely left of centre thought, if you are going south, what about the Isle of Wight?
Lots to see, my mate Paul organises a big bike night once a month at his training centre at Newport Football club when in excess of 500 bikes turn up and it is not that expensive.
Food for thought?