Spotify doesn’t let you keep permanent copies, but if you’re paying a tenner a month you can download the phone app that lets you have offline copies that you can listen to for as long as you keekp paying. There’s also an option to buy (through another site), but I’ve never felt the need to use it. I use it in much the way you suggest, find something I like the sound of, look it up on Spotify and download it to my phone in offline mode to listen to in the car.
Lastfm is good for finding new music similar to the stuff you like. There’s now a Lastfm app within Spotify, so not a lot of point in having them both.