Here's a way to think of tuning.
Take one of those simple tunes (somebody should come out with a whole book of those things!) and really work it over, listening carefully to the intervals. Don't just play it in sequence from beginning to end. Go back and forth. Pause on a lot of notes.
Listen, listen, listen.
Have the tuner on but don't stare at it. Use it to check what your ear is telling you.
Think of an out-of tune melody like a wrinkled shirt. Just as you would work over the wrinkles carefully with an iron, you need to work out those wrinkles in your tuning. Back and forth, be patient. Get everything to line up.
