It's a small island and not a thriving city that offers lots of options. There are many areas which are not allowed to any outside visitors. There are many islands that you can visit like Havelock, Neil, Ross and Viper. But again not every island has options to stay. Though Havelock is a must visit for its white sand beach. Port Blair has the biggest historical attraction, the Cellular Jail where Indian Freedom Fighters were kept under inhuman conditions. It shows an engaging light and sound show everyday. If you are a water sports enthusiast then scuba diving and snorkeling should be considered. Read and research on the web to decide which all places you would like to visit.
While most points are covered by other travelers, I would strongly recommend hiring a two-wheeler on Havelock to explore the interior parts as well. Do not miss Kalapatthar Beach