I am going to go over some steps that can help you learn how to play the piano. These techniques work and will make your journey easier over time.
Relaxation
Try to touch the piano keys lightly and gently. If your fingers are rigid, stiff, with the anxiety of trying to get the right notes at the right time, it will be much more difficult to get it right. A relaxed hand is much easier to get the the right place at the right time.
Enjoyment
Play something you like and that you are familiar with. In time, you will be able to look at any sheet of music and play something close to what is written, but there is no sense adding the burden of playing unfamiliar music on top of learning to press the right key. When we learned to type, we learned by typing familiar words. We didn't learn to type obscure chemical names or technical jargon. The same applies to the piano. When starting out, play what is fun for you. Anything you play will develop your skill.
If you are practicing to play by ear, play what you are familiar with: advertising jingles, TV theme music, popular folk songs, etc.
For improving Fluidity:
Do you play by ear? I recommend you try it. I found that my fluidity went up greatly when I started playing by ear. Or in other words, without any music, sit down and play the melody to some songs that are very familiar to you. Add some harmony as appropriate. This should be music that you have not previously played from a printed page.
Since I started learning to play by ear, the music I play from a printed sheet sounds better. My fingers just seem to anticipate where the next note is going to be, even if my eyes haven't quite caught up. You don't have to become an expert at playing by ear, but if you practice until you can do it a little bit, it will greatly improve your sight reading capabilities!
For Improving Memory:
Another idea that has more to do with biology than anything else, is to play the piece you are learning immediately before going to bed. It should be the last thing you do before laying your head on the pillow. During the day, our brain stores many things in short term memory. These memories are transferred to long term memory during sleep. If you are doing anything after your practice, the memories of your practice session will tend to get overwritten by the events you do later in the day. Playing the song one time just before bed, will bring these memories to the foreground. They will be transferred to long term memory more efficiently.
Keeping the Rhythm
The trick is to play what you play in the proper rhythm, even if you only play a portion of the song. In other words, practice keeping the rhythm going at the expense of everything else. Don't play the cords, or play one or two notes in the cord rather than the whole thing. Before long, you will be able to get more and more notes included in the cords, or you will learn which notes are optional. In most cases nobody but you will realize that there is a note missing.
Where to Put your fingers For scales:
The orthodox technique, for example if you are going up the scale with the right hand, would be to play the first three notes starting with your thumb, and then after playing the note under your middle finger to play the next note with your thumb again. (The thumb crosses under.) Going back down the scale, starting with the pinky on your right hand, you would play five notes, and then your middle finger would cross over the top of the thumb to play the next lower note.
Playing with the left hand is a mirror image of what the right hand does.
For regular music:
The same sort of arrangement applies, but it is also common to just pick up your hands and move them so that they span the next sequence of notes you are expecting to play.
Playing with Both Hands
(Nearly always) play with both hands, don't try to separate them.
Don't worry about skipping a note here or there in the harmony. If you don't get there in time, just go on to the next one, or make up a harmony! Few listeners will be able to tell. After a while your left hand will develop a feel for where the notes might be expected to be. After a while, your right hand will be able to pick out the melody essentially by ear without having to pay so much attention to the notes on the page, thus allowing you to pay more attention to the left hands harmonies. These two trends will combine together to make playing easier and easier. Doing a little bit of playing by ear, and a little bit of sight reading new music, will help to strengthen these trends.
These are just a few techniques that will smoothen your learning experience. Never stop finding ways to make learning the piano easier, and then whatever you learn, share freely to those out there who need a little help as well.