Depending on how clean and detailed you want your models, the amount of tools you use may vary. Starting off with the essentials you'll need some wire cutters and pliers or tweezers.
I got these at Target and you should be able to find something of the sort at any hardware store. The wire cutters are used to cut the pieces off their runners and the pliers are just in case you have large or misshapen fingers that have a tough time dealing with the tiny pieces.
However, if you're willing to spend a little more money and are a bit more serious about your models I highly recommend getting some Tamiya side cutters.
These things are great and probably the cleanest way to cut apart plastic pieces. If you'd like to take things even further you can also use a hobby knife or xacto knife to clean and even out any mistakes. A small set of files are great for that too.
Once your model is smooth and clean you may want to take it one step further and paint it. The paints that I use are Tamiya spray bottles and Tamiya mini acrylic paints for details. Enamel paints are also really great details and certain effects too. I should say I am no where near a paint expert and paint preferences vary from person to person.
Some other odds and ends you may want to use are masking tape and glue or cement. The tape really helps with painting, whether its just holding a piece from moving while you paint it or even to mask off parts of a piece. And of course the glue and cement can be used to cover seams and attach pieces together.
If you're interested in where I got most of this stuff check out the Suppliers page.