A portion of developing pitch has to do with muscle memory. There are certain places in your vocal range that you could feel. Along with this and practicing at the piano, you can over time develop more controlled pitch.
Sing from your lowest pitch to your highest pitch without going into falsetto (where a high break in your voice is--where you're no longer singing at full strength). You can usually tell which is your lowest pitch because it is the one right before your voice goes into what is called a vocal fry with no discernible pitch.
Experiment with your voice. Go above full strength. Sing in falsetto. See how high you go. When you start to tire, go ahead and stop.
Make a daily routine of pitch building with a keyboard, piano or other instrument, essentially repeating steps one and two. The repetition will strengthen, stretch and continually warm up your voice. Your voice will gain endurance.
Slow down your singing. As you sing each pitch, make a mental note of where your vocal chord is at. You can do this just by stopping on a pitch and holding it and make an estimate about where your vocal chord and larynx is at. You can even put a finger to your throat and get an idea of where your vocal muscles are at.
Keep practicing, holding each note as you go up your range, each time making a note of where your your vocal muscles are at. You will soon figure out where your muscles should approximately be at given pitches that you sing.
The technique within the voice (or any other instrument) will make it easy to keep yourself on the right notes. It will also help fine-tune your ear into recognizing pitches so that your pitch becomes almost impeccable!
Developing muscle memory is also important for piano, guitar, and many other instruments as well. Music requires a lot of motor skills in order to play or sing.
I know that the steps seem simple to develop this skill, but it takes actually a lot of practice, preparation and effort to master it. The good news is, if you are a musician or are planning on becoming one, you will have already adapted to the skill of practicing, so developing those chops won't seem like a chore! I hope to expand on this topic in some upcoming pages. Feel free to contact me if you're interested in learning more about this or if you have any questions about developing your musical chops.