The main problem with learning piano independently is that students tend to do what feels natural to them, not what gets the best results. Posture is an important part of playing the piano. Without a teacher, you may unintentionally find your posture slipping, even if you've researched the proper way to sit. Your back must be straight, and your stool or bench must be properly adjusted for your height. Both feet must touch the floor and your forearms must be parallel to the ground.
Without instruction, aspiring pianists typically use their strongest fingers when they start out: the thumb, index and middle finger. While you may learn simple tunes quickly, you'll run into hurdles when you get to more intricate passages. Buy music and exercise books that have all the fingering written out. Learn the fingering and follow it closely, even if playing is uncomfortable at first.
Self-taught piano students often want to learn specific pieces that they've heard. They may spend all their time learning these pieces, rather than doing the myriad technical exercises that a piano teacher would assign. If you're going to learn the piano on your own, research exercise books and buy several. Books by Hanon, Pischna and Czerny are considered standard. Learn the main 24 scales and arpeggios. This will help you play those intricate sections you keep messing up. Take advantage of the Internet -- hundreds of free tutorials are available online.
A competent piano teacher shows her students how to read music and understand music theory. Without this guidance, beginners may try to learn music by ear. This approach may speed your progress initially, but you will reach a glass ceiling when memorizing complicated pieces. Music theory is important, especially if you're learning jazz piano, which relies on improvisation around chords. Take extra time to learn about chords and harmonies. Go to your local library and check out books on music theory -- then apply that knowledge to your playing.