I want something that will easily update all my applications. The Mac App Store doesn't do this with applications not purchased from the Mac App Store, I need something else. In addition, I would also like to be able to browse apps.
If you want an app-based app store alternative, then try Bodega. It's a pretty good app that allows user to download/buy apps that existed before the app store did. However, if that's not enough, you can always use different websites such as www.macupdate.com, www.iusethis.com etc.
Check out Homebrew Cask. Homebrew is a package manager for macOS. While originally it provided the facility to installed command line tools not provided or updated by Apple, of late it has been extended to install GUI apps.
Homebrew Cask can automate the management of apps with simple commands and you can also write Bash scripts to automate the task.
You can easily search for packages using
brew search <package_name>. To upgrade all the installed GUI apps, you can simply run
brew cask upgrade.