With the context of the question / questioner, I've rolled these answers into:
The railway line from Salisbury to Westbury doesn't open until 10 O'clock on a Sunday morning, so we can't suggest any sort of connecting service into the Swindon to Weymouth train that First Great Western are running. You can, however, use the South West Trains service via Southampton for a day in Weymouth, leaving Salsibury at 08:20, 09:08 or 10:13, and getting to Weymouth at 10:35, 11:35 or 12:35. Sample Return trains from Weymouth are at 17:48, 17:56 and 18:48. Prices via Southampton are considerably higher - the adult fare is 29.80 (as opposed to the 15.60 fare from Swindon to Weymouth).
It is possible on a SATURDAY to travel from Salisbury to Weymouth, changing at Westbury, for 21.90 (That's offpeak day returns from Salisbury to Warminster at 6.30, and Warminster to Weymouth at 15.60; you can buy all the tickets at Salisbury, and don't need to get off and on at Warminster). The Saturday train at 08:34 connects to Weymouth at 10:57, returning at 17:28 from Weymouth with the connection getting back to Salisbury at 20:32.
Finally, I'm advised that it's possible to do the trip for 19.65, with a train from Salisbury to Yeovil Junction, then a taxi (on Sunday) or connecting bus (other days of the week) to Yeovil Pen Mill to pick up the First Great Western train. The taxi or bus fare is NOT included in the price of this ticket, and it seem peverse to suggest to people who are trying a service out that they should use a taxi to travel between 2 railway stations on opposite sides of an intermediat town. Very clever piece of routing by the person who came up with this answer, though!
Thanks!