First, check online the available flats for rent in the biggest real state portals:
These portals collect entries from mostly all Real State companies websites, and has nice search interfaces in English, so you can browse the Estonian Real State market.
When you see the price of the rent, remember that at the first month you will have (usually) to pay 3 times its price for the Real State Agency: the Rent in Advance; the Deposit; and the Real State fee; So if you want to rent a flat for 5.000 eek, you will have to pay 15.000 when signing the contract.
Check with attention the Heating System of the flat. There are many options in Estonia, going from Oven heated houses (where you have to buy wood to heat it in winter), Electric Heaters (the most expensive system), Gas Heating, Central Heating... Ask always for the prices of the utilities (water, electricity, heating) in summer and winter!
Be aware that this is a very volatile market, so if you saw a interesting place, you have to call the agency, book it for visitation, and in 24h decide if you are taking it. It might happen (often) that you will be racing against other consumers, and the first one to sign and pay everything gets the deal.
Usually you will find information in English, but here is a little glossary to help, just in case:
Kinnisvara - Real State
Üür - Rent
Müük - Sale
Korter - Flat
Maja - House
Korrus - Floor
Tuba - Room (including living room and bedroom)
Magamistuba - Bedroom
Vannituba - Bathroom
Hind - Rent Price
Tagatisraha - Deposit Money (usually the same as the rent)
kommunaalkulud - Utilities (water, electricity, heating)
Well, if you speak Estonian, you might try to find a flat in the newspapers and rent straight from the owner (avoiding paying the Real State company fees), but its not common to find owners that rent to foreigners. Also another option for the young foreigners is to share a flat. Do you have any good hint for living in Estonia? Post it in the comments!