To watch a video in the Youtube app, you simply tap a video thumbnail.
The video will open and instantly start playing. It doesn't open into full screen mode automatically.
Below the video, you can see more details about the video and the channel it's on, as well as comments and related videos.
You can minimize a video and continue to watch it while still browsing the app. To do this, you drag down on the video and it shrinks into the bottom right corner.
You can then get rid of the minimized video by swiping it from right to left.
If you like what you see, you can subscribe to the channel the video is on by tapping on the red "subscribe" button.
Subscribing reveals a new button with a bell icon. As the overlay states, tapping on this new button lets you get notifications about new videos from the channel you just subscribed to.