You haven't make it clear whether you are driving or whether you are getting public transport?
Well I will try to help anyway.
If you are driving, no you do not have to pass through Bristol, you just take the A368.
If you are relying on public transport, there is a Bus known as the AirDecker which goes from the Airport to Bath City Centre making several stops on the way and takes roughly 50 minutes depending on traffic and costs £14 for a single journey.
Alternatively you could get the Bristol Airport Flyer to Bristol Temple Meads and then jump on the train from there to Bath Spa, The Airport flyer takes around 20 mins (again traffic permitting) to BTM and the journey time from there to Bath Spa is 11 minutes. The Airport Flyer cost £7 for a single Journey, a single train trip on to Bath Spa £7.80.
I'd just take the Airdecker. It may take a bit longer but once you get on it at the airport there's no hassle of having to transfer your luggage to the train. Also, the timings for the combined bus/train journey don't take account of the fact that your arrival at the station by bus may well not coincide with the time of a train to Bath.
Sorry - going by public transport.
@Christabel @Dean G That's great guys, thanks a mil!
USed it for the first time yesterday. Very disappointed that there doesn’t appear to be an online guide to ALL the stops. Outline on the screen on the bus but all of a sudden it announces more stops in between. Could have got off earlier if I had known in advance. Takes time but once your on there is no hassle.
That sounds like a fault: I use it regularly (tonight being an example) and have never noticed that, and I know the route well having worked near the airport for 20 years.