What are the basic warmup practices and exercises for piano playing?
Generally, warm ups on the piano are very similar to things you might do on other instruments. Playing through scales, arpeggios, chords, or Hanon exercises are all traditional warm ups. These exercises are basically to get your fingers moving and get the blood flowing.
Other people, in the interest of warming up, will just run the hands under some hot water for a minute or two to heat them up.
Personally, I don't really feel the need to warm up at all. My hands will warm up in a few minutes from playing anything, so I'd rather spend that time actually practicing something. This is not to say that I don't practice scales and arpeggios. I just don't warm up with them.
I don't warm up when I play the piano, but I have several exercises that I go through to help with various parts of my playing.
Playing phrases with runs (a la Chopin). I play the melody in accented staccato, starting slow then getting faster. This helps a lot with accuracy as accurate staccato is a lot harder to play than accurate legato.
To build finger strength, hold down 5 notes with all your fingers on one hand, then raise your 4th finger and play the note, over and over again, without raising the other fingers. Do the same with the pinky, as they are the weakest fingers.
These two are the main two I use.