I've been to all of them except Bagshot and I have lived in Farnborough.
Farnborough is dull as ditchwater. The whole time I was living there it didn't even have a cinema (I hear it has now), the shopping is boring, there is no cultural scene whatsoever, no nightlife. I think it suffers from being too close to London and the assumption that anyone who wants a bit of excitement will be willing to travel there (to be fair, it is only about an hour on the train). The main reason we moved away was increased activity on the airfield which meant the air always stank of aviation fuel. On the plus side there are a few nice green areas along the Blackwater River which means you don't have to walk that far to get out in the countryside.
Farnham is pretty and kind of villagey in character, with one of those very wide old market streets. I like it aesthetically, but I wouldn't have been happy there in the long term as I do prefer more of a city vibe (this may not be an issue for you).
Aldershot is best known as an army town but the town centre isn't bad (or wasn't when I lived in the area) and it felt a bit more "alive" to me than Farnborough.
Camberley is like Aldershot without the army.
Basingstoke would be my pick, mainly because it's bigger, but as I said, I am a city girl at heart.
If you like small places I'd suggest also having a look at Fleet. I always rather liked it visually, it has fairly interesting shops for its size, a lovely nature reserve at Fleet Pond, and it isn't much further away from London than Farnborough by train (next stop along on the direct line).
I also liked Woking (where I used to work) but it might be the sort of bigger place you are trying to get away from.