South to Sary Tash, Tajikistan and Irkeshtam
Take marshutka number 305 (no more than 1 dollar) from the Baazar. They leave every half hour during the day and takes you onto the last village in Osh province on the road to Sary Tash (M41). Note that bus turns off the main road at the last village, so make sure you get off before it turns off. From here it should be easy to find a truck to Sary Tash taking roughly 3 hours in summer. Some drivers do ask for money, if so, a reasonable price should be 200 som.
If hitching to Irkeshtam, note that it is near impossible to leave Osh and to cross the border on the same day. There are places to stay in Sary Tash and it is always possible to pay a local for a place to stay in Irkeshtam (200 som is reasonable).
North to Uzgen, Jalal-Abad, Bishkek
Marshrutka #121 leaves from the intersection where E007 meets Maliev street (500m south of the intersection from which minivans leave for Bishkek). It takes a left at the large roundabout 4km later. Ask the driver to get off at the intersection with the main road to Bishkek 2km later. Fare is 10som. Just start walking down the road and look at the drivers when they walk past. (It was very easy to get a ride and the road is in great condition)
Pro-tip: download the 2GIS app, it has offline maps with bus routes for a lot of main cities. It's likely there's another marshrutka plying a similar route from wherever you're staying.
Places to visit
Climb up ot St. Mt. Suleyman located right in the center of city.