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If you want to go trekking, CBT Osh is the closest thing you'll find to a tourism office. It is a private company, but they do offer free information (up to 10 minutes) if you go to their office (which is also a guesthouse). Their main activity is offering guided trekking tours throughout the country. Keep in mind that they have an incentive to sell you their more expensive tours.
A mountain looming over the city full of ancient sites of worship, some still in use, dating back several millennia. It is considered the most complete example of a Central Asian sacred mountain, and accordingly has been inscribed as a UNESCO World Heritage Site. A cave in the mountain is the site of a museum containing a collection of archaeological, geological and historical finds and information about local flora and fauna. Also here is a very compact mosque/mausoleum tucked into the mountain called Dom Babura Mosque at the other end of the prominence. Overall it's a good place to have a hike and take in the views.
Mosque of Ravat Abdullakhan / Ravvat Abdullakhan Mechiti / Равват Абдуллахан мечити
Said to be oldest mosque in Osh, a tribute to 16th-century Rabat Abdul Khan. Non-Muslims most likely not allowed entry.
Basic collection of Soviet realist pieces along with some regional Kyrgyz works.
Mavzoley Asaf Ibn Burkhiya
Interesting Islamic architecture at base of Mount Sulaiman, named after King Suleiman’s vizier.
Osh New Mosque
A mosque of notable statue with blue mosaic tiles that you can see before going up Mt Sulaiman. Tourists not really allowed in though.
Blue and white church with a stone gate in front stemming from the early 20th century that is supposedly the only Russian church left in Osh after the demise of Soviet Union and events that followed.