You can find a very good selection of local and international brands for both soft and hard alcohol from most supermarkets but if you'd like to get some good advice and the chance to buy something more unique, the smaller specialized stores are your best choice.
Estonia's flagship alcohol brand is the rum-based, herbal infusion liquor Vana Tallinn ('old Tallinn'), produced by Liviko, and it's a popular pick as a local souvenir to bring back from your trip here. There are three3 strengths: 40,45,50% and several different cream variations also. 

Tartu's own A Le Coq beer factory is certainly worth a pesonal visit if you're a beer lover. They have a gift shop and also offer tours of the factory (both closed for renovation till August 2021).

Please note that alcohol sales are prohibited from 22:00 until 10:00.