The way to access the drop down menu varies, depending upon your device type. This post shows how to access the drop down menu on various Palm devices.

To get a new pet, you will need to give your current pet up for adoption (from the drop down menu) and then adopt a new pet. I like to keep looking until I find an unusual pet with mismatched body parts (Arthur calls them mutants - look here). They are often very cute and quirky.

You might find this discussion about the kind of pet that you get in the game helpful.