There are many coastal and countryside walks from the home without the need for a car, and many more if you don't mind a small drive first. The park has a small outdoor swimming pool and small shop, and is within easy access by car to many cafes, restaurants, museums and supermarkets. At about 6pm most days you can watch the cruise ships begin their journey through the Solent from Southampton, which is within easy reach by car.

The Solent is a hive of activity with every type of watersport you can think of - https://www.visit-hampshire.co.uk/things-to-do/activities-and-sport/watersports

Portsmouth and its Historic Dockyard, D-Day Museum & Overlord Embroidery, Spinnaker Tower, Submarine Museum (Gosport), etc. are within easy reach by car.
https://www.historicdockyard.co.uk/
https://www.nmrn.org.uk/plan-your-visit-royal-navy-submarine-museum
https://theddaystory.com/
https://theddaystory.com/discover/overlord-embroidery/

Southampton and area has numerous points of interest - https://www.planetware.com/tourist-attractions-/southampton-eng-hmp-so.htm

Within easy reach by car is the New Forest National Park, within which is the Beaulieu Motor Musem and its attractions - https://www.beaulieu.co.uk/
https://www.newforestnpa.gov.uk/

and the South Downs National Park - https://www.southdowns.gov.uk/our-history/why-are-we-a-national-park/

Hampshire has 21 castles you can visit, and Stately Homes, including Highclere Castle where Downton Abbey was filmed.
https://www.visit-hampshire.co.uk/
https://www.greatbritishlife.co.uk/homes-and-gardens/places-to-live/how-many-castles-are-there-in-hampshire-7287378
https://www.highclerecastle.co.uk/

You can also take either a car ferry, passenger ferry or even "fly" on a hovercraft over to the Isle of Wight, either for the day or longer.
https://www.isleofwightguru.co.uk/isle-of-wight-ferry-beginners-guide.html
https://www.wightlink.co.uk/
https://www.redfunnel.co.uk/en/
https://www.hovertravel.co.uk/