I live in Fleet so know the area well. To get between those two places the A30 is the obvious choice and the alternatives do involve a lot of extra miles. The A30 is generally fine in that area, quite built up from Bagshot to the big roundabout at Blackwater where there is a large Tesco and M&S. The only stretch to be a bit careful of is the arrow straight section past Blackbushe airport, it's not a very pleasant 2 miles and is the only road locally that I am wary of due to the cyclist accidents that happen along it. It will be busy enough along that section on a Sunday morning due to the very big market at Blackbushe airport so don't expect it to be quiet, but once that section is done the A30 opens out a bit and is a much nicer road.
If you are interested then I have some very scenic country roads to take you from Hook down to New Alresford just outside Winchester, you can do that stage of the journey without touching a main road and it is one of my favourite rides locally. This sort of thing
http://goo.gl/maps/XgpTe which IMO is a better bet than using the A30 and A33 to Winchester.