Dear PianoStreet Forum Members,
This is my first time posting here, so please bear with me. I joined this community to ask this question and see if I could gather some useful ideas. Thank you for checking out my topic, if you have some advice, I would love to hear it

I've been playing the piano for about three years now. I'm very embarrassed of my progress, seeing that I haven't come a long ways. Some of the stuff I have learned has been:
Minor Chords
Major Chords
Basic Songs (like Favorite Things, Bach's Minute, Let It Snow, etc.)
I guess the part that I'm most ashamed of, is that I can't really play with two hands. I did learn how to do chords with my left hand while I was playing the melody with my right hand for the folk song, Clementine, but it didn't sound very smooth. I love listening to music, but I have trouble playing-by-ear.
I would like to be able to play some longer and faster songs. I also want to be able to play with both hands. Honestly, I don't know where to start. I watched some videos on Youtube of people's whose hands fly across the keyboard and I wish I could do the same. I know it will take a lot of practice, and I'm willing to spend the time, but I would appreciate some pointers on where to begin.
Also, keep in mind that I haven't had a piano teacher, but have been learning by myself due to a tight budget.
I thought about posting this question on Yahoo! Answers or the like, but realized that I would most likely get a much more thorough and helpful answer on a dedicated forum such as this.
Thank you for reading this, I'm looking forward to your help and advice . . .