The closest large supermarket is Asda, however if you are thinking about coming to Weymouth in July/August/September be aware of the Olympics.

The UK has decided that, despite assurances from all in the know, sailing IS a spectator sport and so has basically crucified Weymouth with a couple of years of constant road works (still ongoing).
For the Olympics, areas of Weymouth have been separated as ticketed viewing areas and there are plans to create a large free live site, presumable with a large screen display of the sailing, on the esplanade somewhere.

The rumoured plans are:
No public parking in town car parks. Weymouth out of town car parks are being given over to the London Olympic committee who have raised the charges to about a tenner a day with a large booking fee. Other further away park and ride car parks are being created to enable the people who would visit Weymouth to still come, however I can't recollect if there has been any mention of different prices for them.

So as we understand it, because of the security issues, there is NO parking in town, large areas will be no go for vehicles and all coaches will drop at the swannery car park with planned guided human crocodiles to deliver people to the various areas in town!

It could be different on the day but then, as locals, we are the last to know the actual facts!