At first you are most probably going to need lots of items, such as kitchen hardware, bed linen or clothes. These first purchases may turn to drain some of your first month's budget, so you could consider taking a look at some of the second hand shops that are spread all over Tartu. Several of them sell all sorts of items, from clothes to microwaves, so they could be of help while you settle down.
There are also groups on Facebook where both locals and international students sell their items at a very low price. For a start, you could try here:
This shop established in Tartu sells used and new bikes and you can take a look at their online catalogue.