Are there any finger exercises I can perform even when I am not near a piano to improve my dexterity and strength of my fingers?
An exercise that I was taught, have seen many times, and also have used with others. It can be done anywhere. Instead of tapping my fingers, I do this exercise.
Place your hand on the table (or whatever surface) as though you were resting your fingers on the keys. "Play" the sequence 1-3-5-2-4, repeating it over and over. Things to work on are your rhythm, make sure that each "note" you play is even. (For instance, 1-3--2 are more dominant fingers, but make sure ---2-4 are doing their share of the work!) Work on being able to keep that up for several minutes, keeping the 1-3-5-2-4 progression intact. I find myself lapsing into 1-3-5-43-5-1 or something equally funky if I'm not careful.
The temptation here is to do this with your dominant hand, but make sure you also do it with the other hand. The tempo will be much (much, much, much...for me) slower, so spend more timing working with that hand.
This should also be done with both hands, keeping each note in sync. For me, it usually ends in disaster. (:
I have used the various table-exercises mentioned above. I suggest getting the Hanon The Virtuoso Pianist in Exercises and the Schmitt book on exercises.
These will give you plenty of patterns to choose from as well as some good stuff do while at the piano.
Now keep in mind that it is more fatiguing to play on a hard surface such as a table, and with fatigue you can get muscle stiffness.
Having said this, you'll find that if you do the above exercises carefully and with relaxed arms and hands while at the piano, for maybe 20 minutes per day, there's no need to do table playing unless you're going to be away from the piano for quite some length of time.
Keep in mind what you are really strengthening are the tendons as there are no "muscles" in the fingers. All the work is really done in your forearms and biceps and the tendons are pulled over your elbows by your finger movements.
Book+video: Seymore Fink, Mastering Piano Technique (search)
It has good full arm exercises, to practise both with and away from the piano. The idea is to work out the muscles involved in playing piano - from shoulder to finger tips, and groove the patterns involved in playing, like the circular motion while playing an arpeggio.