Cycle touring in Hampshire is probably best on the Isle of Wight. There are some other isolated routes too, such as National Cycle Network Route 23 between Basingstoke and Winchester and National Cycle Network Route 24 between Salisbury and Southampton, but these routes contain long stretches of rural cycling with not much to see and do on the way. Do not be fooled by the famous South Downs Way. This route is really something for mountain bikes only and not suitable for hybrid bike and/or family "cruising".
A great help for you to plan your routes in Hampshire will be the Sustrans' Hampshire & Island of Wight Cycle Map. This map covers the National Cycle Network of Hampshire. For cycling on the Isle of Wight, we recommend using this map too, in combination with our route suggestions on our Isle of Wight page.
If you are interested in traffic-calmed long-distance cycling on England's South Coast, please note that National Cycle Network Route 2 is far from completed. This route, supposed to connect Dover with Cornwall, has various missing sections. You'll end up cycling on busy main roads for considerable distances and also have to plan and research your way to make up for the missing gaps.
If you are after a continuous east-west cycle route across southern England, consider our London-Land's End Cycle Route Book , which also features a route from Dover to London, thus providing the Dover-Cornwall connection. It comes in a handy spiral-bound format and GPS-tracks are free when ordered via our own websites.