The cliffs above are in West Bay which is just over a mile from Bridport centre. It also has a nice old harbour and lots of little food huts (Rachel's is recommended).
Other places worth a visit if you have time:
Lyme Regis (5 miles)
Lyme Regis is about 5 miles from Bridport and is an old harbour town famous for fossils and the Cobb harbour. Definitely worth a trip.
Abbotsbury (10 miles)
Abbotsbury is a small village on the way through to Weymouth. Traditional stone & thatch cottages with a couple of nice pubs, an old monastery up a massive hill (with views of Chesil Beach) and the country's only (and best) swannery, featuring loads of swans which you can feed or just look at.
Lulworth / Durdle Door (30 miles)
Durdle Door is a big rock thing in the sea. It's very nice but there's a lot of steps down (and therefore back up) - so make sure you're up to it!
Dorchester (15 miles)
The county town of Dorset. Quite a small town but worth stopping by if on the way to Weymouth.
Weymouth (20 miles)
Weymouth is probably the biggest town nearby. The harbour is worth the visit and there's also the old fort on the hill behind with a view over to Portland. A little further across Portland is the Portland Bill lighthouse which was famous from some puppets a while back.
Beer (22 miles)
Another nice seaside town. You can go have a beer in beer before you head to the town of Ham for dinner.
There's way more places worth seeing than listed here as well, so it's worth having a look online.