Taxi services can take you and wait a few hours for about 1000 som ($25). From the newly renovated lodge you can trek to the stone house located at 3300m and it takes about 4-5 hours.
English speaking taxi service is available through Iron Horse Nomads for 900 som, call +996-703-751-844 for their dispatch.
This gorge is rugged but very accessible from Bishkek (the main gate is just 30km away), and you can sit by a waterfall all day, hike up to a glacier and even ski on it (even in the summer), or trek to the regions highest peaks. The majority of the canyon is found within a state nature park and there's an entrance fee of 60 som for those on foot or 120 som if you're in a car. In summer there are recreational facilities, baths and a sauna. From the lodge you can climb Corona Peak or Uchitel peak 4585m or head to glacier.
The best thing to do is bring your own tent and sleeping bag. The only all year accommodation is a wooden hotel (460 som) with a dozen uninspiring double rooms with a shared bathroom and no running water, and there's a mediocre breakfast thrown in for an extra 80 som.
A better option is the nearby A-frame alpine lodge but this is much more expensive. Try to avoid Saturday or Sunday when locals swamp the place and sit around consuming large quantities of vodka.