As long as you bring your cruise card with you – which would be tough to not bring as ships require you to sign off the ship when you debark, and another form of ID like a driver’s license you can come and go as you please on cruise ship as many times as you would like in port. Keep in mind that if you are tendered you will have to keep taking the tenders back and forth and they do take some time.
As for what else to pack – bring what you think you will need. If you are going to a beach then pack as though you were going to a beach at home – bring a towel, sunscreen, sunglasses, bathing suit, etc. (Actually I would wear my bathing suit off the ship.) If you are going shopping and you have never been to that port before you may want to bring a shopping map – which the ship will provide, your wallet, etc. If you are docked and want to go shopping for the morning then to a beach I think it’s easier to just leave in the morning to shop then come back to the ship to drop off my packages and change for the beach. (Or vice-versa if you want to go to the beach first.)