Tashkent is the capital of Uzbekistan.
You can follow this guide or you can try to use the app 2GIS (free for every phone) and find a bus or marshrutka going out of the city to the direction you need to go. It's very useful to find the best and easiest conection to get out of Tashkent from the center.
North towards Shymkent
The border crossing is just a few kilometres north of the city and it's a pedestrian-only crossing. This means only taxis and buses will go there. It's probably the best idea to go there by taxi or bus i.e. from Yunusabad bazaar. The border crossing might be crowded and it can easily take 2 hours or more. After it you will be surrounded by taxi drivers. Just ignore them and follow the road straight on. You can try to hitch some of the few local cars or walk about 4 km to the main road to Shymkent which is a dual carriageway with some more traffic.
SouthWest towards Samarqand
Take the metro to Olmazor station (last station on Red Line). Then take the tramway or any bus going South on the main road, or you can also walk (30 minutes), to the Ring Road. Start hitch-hiking just after the Police checkpoint.
East towards Fergana and Kyrgyzstan
Tricky! Don't trust the maps in lonely planet of Tashkent, whice shows a good spot by Tashkent station. Take a bus 110 direction Angren. You can either take it just to the highway or all the way to Angren. When on the highway you can hitch to the checkpoint, where the police will stop a car for you to Angren or further..
Ask your driver to drop you off at a metro station.