Bishkek is the capital of Kyrgyzstan
North to Kazakhstan
The main road towards the border from Bishkek runs north from the Alamedin Bazaar, but nearly all drivers stopping along this road are going only to local villages and, what is more, demand money. An easy option is to take a a cheap shared taxi from the Eastern Bus Station to Korday, the Kazakh town on the border. The shared taxi will drop you off before the border, which you can then cross on foot. From Korday, you can easily hitch on to Shymkent or Almaty, or to other directions in Kazakhstan through those cities.
East to Issyk-kul lake
The main road towards the Issyk-kul lake from Bishkek runs east from the Alamedin Bazaar. Take bus to Kant (Kyrgyz: Кант) then walk outside of town and start hitching.