Even if you've only got one day, Tallinn city centre is pretty small. Walking is the best way to get around. Walk up the cobbled Pikk jalg (Long Leg) or the steps, Lühike jalg (Short Leg) to the Upper Town (also known as Toompea or Cathedral Hill), and you can look out over the city and the sea from various terraces. There are one or two open-air cafés up there, so you can refresh yourself. Also the Cathedral, the Russian Orthodox Cathedral and the pink (!) parliament building.
There are plenty of souvenir shops on the Main Square (Raekoja plats) in the Lower Town. As well as those outdoor restaurants that I mentioned before.
There are one or two central churches worth peeping inside, such as Püha vaim (Holy Spirit), Oleviste (Saint Olaf's) and Niguliste (Saint Nicholas'). The last of these is deconsecrated and doubles up as a concert hall, but the mediæval frieze, with Low German comments written on it, is worth looking at. Tallinn (under its old name Reval) was a Hanseatic port, hence the use of the Low German language, which was the language of sea trade in those days.
There is an art museum on Vabaduse väljak (Freedom Square) and right next to it a restaurant in a cellar called Kuku, which doubles up in the evenings as the hangout for artists and writers. During the day, the restaurant / club is open to everyone. Another art museum is in the Upper Town. There are also a number of private galleries, such as Vaal, dotted around the Old Town.
The Kumu gallery does modern art; it's a short tram ride from the centre near Kadriorg Park, where the President's Palace and the tiny Peter the Great Museum are also situated.
Estonia's most notable contemporary artists are (off the top of my head, and according to my tastes) Jüri Arrak, Toomas Vint, Epp Kokamägi, Reti Saks, plus older artists such as Laikmaa, Raud and Wiiralt from the early 20th century.
If you go to the music shop at the foot of Pikk jalg, you can buy music on CD by Arvo Pärt, Veljo Tormis and one or two other contemporary composers.
Stick to the Old Town in Tallinn if you've only got a few hours; you won't regret it. The Wiki entry for Tallinn is here; plenty of photos there:
http://en.wikipedia.org/wiki/Tallinn
The reason I reply in detail is because I've been doing virtually nothing but translating Estonian literature these past few years and have visited the country on several occasions, including a stay in Tallinn of over a year in the mid-1990s.
Two quite different views:
