I'm not obsessed with my range, I actually mostly care about just singing in tune.

If you want to sing in tune, I highly recommend you to learn the solfeggio. I am not going to be too technical on this; it is basically just your do-re-mi-fa-so-la-ti-do, where by using your 'do' as your reference note, you will accordingly learn to sing the other notes relative to it. Do interval exercises starting from seconds and thirds and moving to perfect fourths, perfect fifths as well as sixths and sevenths. The purpose of these exercises is to improve your sensitivity towards intervals when you hear a song, and accordingly you can match the notes correctly. To me solfeggio is a very powerful tool because not only that it helps you to sing melodies in tune, but you will also be able to develop and master your musicianship, since accordingly you can use it as your baseline to add harmonies and chords to your songs.

Speaking of Internet coaches who can teach you to sing in tune, I highly recommend Phil Moufarrege from http://grow-the-voice.com - check out his YouTube Channel. I have seen videos of his students growing to become way better singers despite starting out with severe pitch issues. Not only he made them pitch sensitive, but he also taught them useful vocal techniques that helped them artistically as singers. Others that are useful too include Kerri Ho, Felicia Ricci, Ken Tamplin Vocal Academy, Marnell Sample, and Justin Stoney from New York Vocal Coaching - they are all very good vocal teachers. All the best!

/r/singing Thread