I went to school there! Finally, I have specialist knowledge.
Traffic IS horrid, but once you know your way about you can nip in and out pretty easily, and the park and ride is swish.
In terms of schools. Two grammars: Bishops is the boys and South Wilts the girls, as someone has mentioned. I went to South Wilts (we shared some 6th form lessons with Bishops) and both are lovely. South Wilts has an all-weather running track, Bishops is great for rugby.
South Wilts is on the outskirts of Salisbury proper, and has some good grounds, a field etc, and when I was there (left in 2006) was doing really well. Bishops is right by the cathedral close so very central. It's less relaxed than Swilts, but a fun place to be.
There are also state schools: St.Edmunds (girls) and Wyvern (boys) which are next to each other. Their results aren't as good as the grammars but I have visited St Edmunds and it looks like a nice school.
Both schools are ... well from my experience, very academic, fast-moving and ambitious. Teachers are pretty idiosyncratic, from what I remember, and there is a LOT of opportunity to play musical instruments, do sport, drama, etc. Great schools.
Otherwise there's Downton, the other comp. I know nothing about them apart from they're not great at athletics. 