Let me tell you a story. Years ago, I taught in a music school. On Tuesdays a vocal teacher did lessons next to my room, and one of her students had issues with pitchiness. My piano student would wince every time the vocalist hit a wrong note. The piano student hadn’t studied voice, yet her experience playing piano helped her to hear when the notes sounded wrong.
This means that you can learn to sing perfectly in tune all the time with touchstone instruments. However, it won’t happen overnight after taking a few vocal lessons. It takes years of practice.
If you want to sing classical or pop, the piano can help you sing melodies in tune:
Learning harmony will be more difficult because we don’t have touchstone instruments for that… at least until now. This link will take you to posts with specially tuned sing-along soundtracks that’ll act as touchstone instruments, helping you learn how to harmonize in tune: