Cobham - Pompey on the A3?
Cobham - Guildford iirc parts of it have a cycle path parts of it don't, generally the cycle 'way' follows the old portsmouth road rather than the modern A3 on these stretches. not fun, at times unpleasant but do-able.
guildford south on A3... gawdawful by bike imo, better to go Guildford, Godalming, Haslemere, Liphook, Peterfield. Lumpy but quite nice.
Peterfield south country lanes, to havant and beyond.