Again Tin answer is very good.
Traffic along the Cote d'azure is always terrible this time of Year. Even on the a8 motorway,, its dodgy.
There is some options, possibly the quickest way to get to st tropez is by the ferry. It departs Nice Port, You can take the bus from nice airport to gare routiere ( which is now basically nice city centre as they are demolishing it) and you take a bus from the city centre around place massena to the port.. or you could walk in half an hour, but with bags etc and i child, its a trek. The bus service is very cheap in nice, i think it 4-6 euro to the centre by bus, and all other routes in Alpes maritimes are one euro. You could also take a taxi to the port.. around 50-75 euro depending on traffic and you are lucky with the lights on the promenade:P
The Boat takes 2 hours ( sometimes a little longer, often yacht traffic in and out of cannes) or so to st tropez, and stops in Cannes and Ile ste maugerite.
I think another option is to take a bus to cannes and take the same boat from there.
There is a train to st raphael from nice st augustine, which is near the airport. and take a bus to st tropez, I havent used this before, so not sure about the details, but the train should be around 30-40 euro and a bus is cheap.
Taxi, you can Pay anything to st tropez, it all depends on traffic..
I think the best option, possibly even the cheapest is to rent a car from Nice airport for a day, and the same on your return. make sure when you book, the company has a returns office in st tropez.
I recomend Sixt, they are reliable and honest, we always use them for our holidays. and they have an office in st tropez. The drive as tin said is scenic, or you can take the motorway which is quicker. it has one toll to st tropez around 3 euros.
There is some options, basically baking up tins answer. Hope it helps..