Get the train.
Direct bus transfer from Temple Meads.
Agreed. Log onto the trainline website and buy a ticket that covers the train and bus.
We travel from Bristol a lot and used to drive to the airport, but it is a pain in the arse going through the city centre. The kid was also sick in the car last time so now we go by train.
It works out about the same on the train / bus as a weeks worth of silver zone parking does. Bus drops you right outside the terminal entrance as well.
One word of advice though. Try to avoid Templemeads station on a Saturday night. Not a nice place to be with tired kids and a whinging wife. Full of piss heads and gobshites.