If you are looking to stay to a budget, stay at Camping at Kingfisher Caravan Park, I stayed here for 2 nights, from the Friday to Sunday, and absolutely loved it. We booked a tent pitch that was £20 a night, a total bargain! The site is quite big offering pitches for tents, campervans, motorhomes, caravans, etc but it also hires out static caravans.
We managed to get a pitch next to the shower/toilet block by the trees so didn’t have far to go when we felt the need, although people going in the middle of the night and letting the door slam shut wasn’t great.
The shower/toilet blocks were kept really clean the whole time we were there, it was warm in the night as well. The showers were those push button showers so they turn themselves off after a few minutes and the temperature went up cold and hot if there was more than one shower going but it was ok if you are the only one in there.
They have a club house with a bar, restaurant, an arcade and a few nice outdoor seating areas. They put on quiz nights, BBQs, discos and do raffles, they did a music quiz on our first night and the second night was an Easter raffle and a disco. Both nights there was a BBQ in the outdoor seating area out the back by the restaurant.
There is free wifi but it is only in the club house and you need to get the code from a sign at the bar.
The site is close to an awesome beach, 5 minutes to an Asda and just under 10 minutes to the ferry port to get to Portsmouth.
Gosport is mainly a residence area but there is one big main high street with lots of different shops and cafes that lead down to the harbour where you can get the Gosport Ferry to Portsmouth. It is the perfect place to have as a base for going to Portsmouth but Gosport also has lots of water themed activities like the Royal
Navy Submarine Museum, Museum of Diving , Explosion! - Museum of Naval Firepower and there are waterfront trails. If you get a pass for the Historic Dockyard in Portsmouth, some of them cover things to do in Gosport as well.
There isn't a lot of parking so if you can walk or get the bus there, it would be a bit less stressful for you.