I keep seeing photos of The Trundle on FB, glorious poppies.
West Wittering beach you need to book parking now and it sells out.
Day trip from Southsea to the IOW on the hovercraft is good too.
Portchester Castle near Portsmouth is good for a wander round, also the Royal Armouries at Fort Nelson on Portsdown Hill above Fareham. It's worth driving along Portsdown Hill Road just for the views, lots of viewpoints.