Google has made some changes to the web version of Play Music today which makes playlists a little easier to access, by way of adding in a new right side panel that you can slide out by hitting the new playlist button located in the top bar of the web interface. This might seem like a small change, but it will surely make accessing your playlists that much easier. Users still have the option to access playlists through the "Music Library" section that's located in the hamburger menu list on the left, and this will still provide a more visually appealing view of the playlists as the album art thumbnails are located here but not in the Playlist quick access.
Having said that, there are benefits to utilizing the new Playlist button and side panel instead of the Playlists from the Music Library section, most notably the much faster way to create a new playlist. When you open up the new playlist panel, there is a button to create a new playlist at the very top of the panel. It's simple, and it takes just two clicks to get to it. If you access the Music Library section where Playlists is the first tab, there is no way to create a new Playlist from here.
You can create new playlists when you find songs that you would like to add to one, but this takes more time and while it may not happen often, if you're trying to create a playlist quickly or in a rush, chances are you won't want to spend those extra few seconds to look for songs first before creating the actual playlist to add them to. If you never use playlists then these may not be sweeping changes that you look forward to, but for the users that access them frequently this will be a pretty beneficial change as efficiency and convenience are usually a good thing. These changes should be live and showing up for everyone at this point, and if they are there you will see a new popup message in the top right corner where the new playlist button is located alerting you to these changes. If that does not show up then the playlist button likely hasn't been added yet. If you haven't checked out Play Music yet, as a new subscriber you should be able to trial the service for four months as Google extended it past 30 days back in July.