Driving is the best means of getting around Estonia. The roads are good, traffic is light and nothing's very far from anything else. There are major rental agencies in Tallinn. You'll need a passport and either an International Driving Permit or, if a European citizen, a regular licence. Drive on the right.
If you don't mind taking your time to get around, Estonia has good bus and train services. Since the terrain is flat, bicycling is another excellent way to see the countryside, at least in summer. The islands are especially good for cycling.
Domestic air travel is expensive. Baltic Aeroservis links Tallinn with the island destinations of Kuressaare and Kärdla, while charter flights operate between Tartu and Tallinn.
You can get around the larger towns and cities via a variety of trams and buses, although both get frighteningly crowded. Taxis are plentiful and cheap.