I've never paid for Spotify. I started with Grooveshark (as dubious as LimeWire) and moved onto Rdio before it was slaughtered by Spotify when it launched in the united states. I'll never forgive that.
Pandora - Yes, it still exists. Yes, it's radio stations are still far better than any other streaming services. I pay for ad-free stations and use these for music discovery. I listen to stations or full albums. I don't give a fuck about playlists.
Bandcamp - This is where I buy digital albums when I can (iTunes/Apple Music when I can't). Bandcamp allows you to stream songs a few times for free before asking you to pay so I preview albums with this service a lot. You can download music you buy to the Bandcamp app on your phone, but you'll need to download on a PC and transfer to your phone if you want the files in the stock music app (I believe that's only an iPhone limitation)
iTunes is still a thing but called the Apple Music App now. It supports all its old functions but kinda worse all around because it pushes an Apple Music subscription at every turn. Like seiji points out, you can "turn off" Apple Music (streaming) and use the iOS and MacOS Apple Music app as just an MP3 player. It's almost old school iTunes with all the playlist creating tools you could want.
Yes, you can still rip CDs! As long as you have a CD reader...
Longplay - MP3 player app for iOS that only allows you to listen to albums in full. No picking and choosing tracks! I almost exclusively use this for listening to music downloaded to my phone.
Vox - Minimal music player for Mac and iOS that also plays lossless audio codecs. Feels like iTunes with all the bloat stripped out. It's completely free. They have a premium subscription that will sync your collection to the cloud (but they don't shove it down your throat). Also supports Last.fm scrobbling.