How to set the YouTube app to play 360p instead of 240p video?
The YouTube quality settings are determined by each video (although if you select a higher quality, it will default to that for the next time).  Hit the three-dot menu button:



Then the Quality cog button:



Then you can select the quality setting.



Other videos might have more options for the quality setting:



This completely depends on external factors like, for example, the size of the source video that was uploaded to YouTube.

You can also (from the 3-dot menu on the app Home) go to Settings &rarr; General to turn off the "Limit mobile data usage" option, which will allow videos to default to higher quality when you are not on Wi-Fi.