The public transportation network of Tallinn consists of bus, tram and trolleybus services. They are all in an uniform ticket system.
Tickets for a single journey can be bought from the driver (1.60 EUR, no changes permitted). The best way to pay however is to use the pay-as-you-go smartcard called Ühiskaart. The card costs 2 EUR; a ticket valid for an hour 1.10 EUR, a day (24 h) ticket 3.00 EUR, a 5 day (120 h) ticket 6.00 EUR. The validating machine in the vehicle chooses the right fare for you. These cards can be bought and topped up at R-kiosk newsstands and Maxima, Prisma and Rimi supermarkets. For further explanations, see here.
The timetables are here.
Avoid rogue taxi operators, see this warning by the Tallinn airport.
Taxi rates vary a lot. All taxis must have the rates displayed on a yellow sticker on the side rear door window. Reasonable rates are: 2.00-3.50 EUR initial fare, 0.50-0.80 per km fare. Always request a printed request, all taxis must have taximeters and printers.
Some reliable companies are: Tulika Takso (+372 612 000), Tallink Takso (+372 640 8921), Takso 24 (+372 640 8927), Reval Takso (+372 601 4600), Q-takso (+372 644 1525).
You can also use the Tafixy app to order a taxi with your mobile phone (you can see the taxis near you and choose the one you like best).
For further explanations, see here.
A good interactive map providing also public transportation information is offered by the Tallinn city.
Some alternatives are Delfi (Regio) and Google.
Check out the webpages of the Tallinn Tourist Board and the Estonian Tourist Board.
An excellent rough guide with honest reviews is Tallinn in Your Pocket. For in-depth information on Estonia, check Estonica.