You can put your mp3s on your device and play them with the stock music player. The stock player is the same player it's always been, it just has the ability to also play your music from the cloud if you so choose to upload to the cloud.
An example is I purchase some music from google music. It stays in the cloud, but I can stream it through stock Music player (Google Music) along with the mp3s already on my device. At some point, I download my purchased music from the cloud to my PC (so I have my mp3s that I own) and then physically transfer them to my device so I no longer have to stream them, but play them locally on my phone. Some people choose to upload all or most of their music to the cloud so they can keep space on their phone.
If you don't like the stock Music player (Google Music), there are many choices in the market. Power Amp is popular and is the one I like the best, although I'm actually still using "Music" myself. Other popular ones are WinAmp, dB Player, and doubleTwist.
I use iSyncr to sync iTunes playlists to my phone since I already begrudgingly use iTunes for my iPod. It will copy music for you so you don't have to manually manage your music. Some of the other players, like WinAmp, offer the same functionality of autosyncing (just not with iTunes of course).