A good idea is to sing in your car with the audio up high. Then repeat the song at a low volume so your mistakes become more obvious. Any parts/ lines/ techniques you're having trouble with, REPEAT a million times til you can CONSISTENTLY get it right (and I mean, ONLY that part). This is key because singing well has a lot to do with with consistency.
Another trick I do a lot is singing adlibbed harmonies to music. Make up vocal harmonies and sing em along with your favorite songs til you can feel the pitches *click*. This should help with intonation (singing the right pitches) and (DUHHHH!!!!) singing vocal harmony parts.
Practice different vocal techniques like: singing the same note at different volumes (dynamics), *sliding* from one note to another (glissandos), different types/ speeds of vibrato, singing in uncomfortable note rages (i.e. high end or low end of your vocal range), falsetto vocals (if you don't what that is YouTube it) and singing along with a song an octave above or below the true pitch.
Then, of course, it's important to sing without musical accompaniment (a capella). So, sing in the shower! The natural reverb in your shower can help you hear where your intonation problems are.
Be creative. Think critically. Try different things and figure out what helps you improve and what doesn't.