You could do a shoredive from Patong beach, I would not classify this as 'must do' though.
So if I were you, I would do a daytrip to preferably Anemone Reef + 1/2 divesites in the area. IMO, Anemone Reef is the best site of the traditional daytrip sites. The standard program will take you to Shark Point, King Cruiser and Koh Doc Mai, which is also a nice daytrip with lots of variety.
If you have the time (in November it will take you from early morning till late afternoon, in October even longer), you can also consider going to the Similans. Is easily doable on a daytrip, and the diving is excellent.