Go to the southern bus station which is right next to the Adigitrievski Sobor cathedral and from there take a mini-bus 130 to Ivonginsk (Иволгинск). From the last bus stop walk about 300 metres south and you'll get to the highway A165 which is going to Kiakhta.
North East towards Turka
Take bus/Marshutka number 37 from Ulitsa Lenina, on the opposite side of the street than the head of Lenin. Get out at the last stop which is a bit off road in front of the hospital. From there you need to walk back to the bigger street. Just ask for Turka. And don‘t be confused: The bus also goes to the ethnographical museum, which seems to be the last stop - but it isn‘t.
North West towards Irkutsk
Take bus number 129 from Ulitsa Lenina right in front of the head of Lenin. This bus will take you to the „Baikal“ street. This street leads out of town, and it seems to connect villages, so you can easily recognize it. At some point the bus will turn left and leave the „Baikal“ street. Just get out the next stop and walk back 20 meters and stand somewhere beside the „Baikal“ street.