Ol' King George
If you are the sort who likes at statues, and learn of the historical culture of areas, then I guess it's a must that you see the statue of King George III in Weymouth. If you are the sort who likes to see historical monuments ill maintained and covered in bird excretion then your quids in!
Back in the day ol' King George used to regularly visit Weymouth after he had fallen in love with the beautiful seaside resort. He would come and stay in the Prince Regent and enjoy soothing baths in the sea. As a dedication to his fondness of the town a statue was erected in his honour.
In addition a huge White Horse depicting KG riding in to Weymouth is famously located on a hill up towards Dorchester. Unfortunately, in their wisdom, the horse was shown riding out of the town. Madness! (See what I did there? A little pun on the madness of King George. Aren't I just great!.... Don't worry I've got my coat.)

Pub Frontage (And The Other Paying Customer)
The Ferry
Weymouth harbour
View to the Chesil Beach from Portland