You can check the progress of your downloads in [Notifications] > [Downloads].
Play as You Download is available on selected games. This means that you can choose which part of the game you want to download first and start playing before the download is complete.
When you purchase a game, you can remotely start the download to your PS4 from a web browser if your PS4 has Automatic Update switched on.
Your most recently downloaded full games are added to the content launcher. If your purchase is not visible on the content launcher, you can find it in your [Library]. Your Library is arranged with all content grouped together, so if you have purchased DLC, go to the main title to find it.