I'm going to be in Lecce for a few days in Sept and was thinking of relaxing for a day at a nearby beach. Any suggestions of favorite beaches? Also will they have changing rooms or should I wear my swimsuit there? Also is there any problem doing this as a solo traveler? I'd prefer to be with a friend so there would be someone to watch the stuff while the other person goes for a swim.

Ok, beaches in Italy are a bit complicated or just normal depending on where you come from.

Because of history you will find that most beaches have a public and private section. On busy ones the public section tends to be relatively small and depending on local demand can be pretty busy. Resources are limited but there may be an open shower. In the private beach you should have everything, including a bar, lounger etc and places to change.

Before Covid there was a movement to do away with private beaches but I'm guessing that has slowed, other foderites will know more than me on this. I'd use google and google maps to check out "sandy beaches" as some of them can be pretty rocky.

This link gives you some more ideas https://bestinpuglia.com/en/lecce/beaches

You don't say how you are traveling.
Are you looking for an "organized" beach --changing room, chair & umbrella-- or would you prefer a natural beach?

I found the Puglia Guys blog very helpful, and they have quite a few posts on beaches of all sorts--serviced, wild, etc.

https://www.pugliaguys.com/puglia-by-beach/

I'll be using public transit. I'd prefer an organized beach.