Favorite coastal towns near Portsmouth

You are not far from Brighton or Worthing in Portsmouth, both of which are very nice. If you like going in the sea be aware Worthing is a pebble beach. I'm not sure how much you care about that 🫠
There are ferries from Portsmouth to the Isle of Wight which has some very unique seaside areas. As long as you are okay to give up the train for local buses, I'd recommend IoW as it isn't far at all.
There's also Bournemouth and Weymouth to the west, though Bournemouth is closer with a larger beach near the city centre.

Chichester is a lovely city with a glorious cathedral and great Roman history. Close by is the excellent Fishbourne Roman Palace and the nearby village of Bosham is often cited as one of England's loveliest villages. www.thegreatsussexway.org/about-the-area/bosham/

You could then take the train to Brockenhurst in the New Forest and explore the area either on foot, cycle or bus. The nearby villages of Beaulieu and Bucklers Hard are great places to visit, www.thenewforest.co.uk/explore/towns-and-villages/beaulieu/, www.bucklershard.co.uk.

I would then consider a train to Wareham from where you can explore the area of Studland Bay, Corfe Castle, Lulworth Cove and the Jurassic Coast easily by public transport.

We recently made a similar trip by train. Our goal was to explore historic towns and sites; the beach was not part of the attraction. Portsmouth was about one week into our trip and we stayed in a Premier Inn. Our next destination was Hastings. We wanted an apartment there with a washing machine. We ended up in the adjoining town of St. Leonards on the Sea. The town is a bit "gritty" but frankly, we found many of the seaside towns in the area to be a bit worn. The apartment was very affordable, and was located only a short walk from the train. It overlooked Warrior Square Gardens and had a view of the sea. I wouldn't say it was a "favorite" town but it was very convenient for day trips in the area.

If you're considering the area around Portsmouth, here are a couple of trip reports on Chichester and Arundel from two forum regulars who spent some time there last year.

https://community.ricksteves.com/travel-forum/trip-reports/chichester-a-week-off-the-well-trodden-path
https://community.ricksteves.com/travel-forum/trip-reports/3-perfect-days-in-arundel-england

Thanks everyone for the terrific information.

I'll second JC on Chichester. It was a good base to visit the Weald and Downland Museum and each time I've visited the town, I've booked accommodation through the Cathedral's facilities -- https://www.chichestercathedral.org.uk/stay-with-us

I agree with all the recommendations here, you have a lot of great options. Just wanted to add my favourite thing about Chichester Cathedral which is the beautiful stained glass window by Marc Chagall- definitely don't miss if you make it over there. (One of only two places in the UK you can see his stained glass, the other being Tudeley in Kent.)
https://www.chichestercathedral.org.uk/visiting-us/cathedral-plan/delve-deeper-marc-chagall-window