If you're using Chrome, Firefox or Orfox, accessing Messenger online is quite simple. Navigate to messenger.com, tap the three dots in the upper right-hand corner, and select Request desktop site. This will reload the page. From here, you can sign in with Facebook (if necessary) and then go on your merry way.
This website was designed with larger screens in mind, so using this site might be a bit awkward, but from a security standpoint it is preferable to using the app (especially if you're using Orfox).