Bournemouth Gardens are particularly lovely this time of year - there are benches to rest those tired feet and light displays at night. In August the candle light displays are amazing - hundreds of candles in coloured jars transform the gardens. For dates, contact Bournemouth Tourism, Gervis Place.
Bournemouth has a huge sandy beach and a nice prom.
Theres a Wilts & Dorset open-top bus tour from Gervis Place (day-out tickets are cheapest) which goes across on the chain ferry to Swanage, a lovely sea-side resort, where you can also get a steam train ride. The bus also stops at Shell Bay which is particularly spectacular - its best to take your own refreshments. If taking a car across on the chain ferry at Poole, its best to get there very early as there can be long queus. You can go as foot passengers also.
There are many bars, cafes & restaurants in Bournemouth - too numerous to mention, so there is plenty of choice.
BIC - Bournemouth International Centre has live shows and musicals throughout the season.