I'm looking for complete freedom when I'm playing. I hate constantly being stuck in one key. One of the best sounding ways I've found to modulate from key to key is by playing a predominant diatonic chord to a dominant chord of the target tonic to the target tonic. Is there any way to memorize which chords are the predominant, or even dominant, for other keys? Before I just sit down, write out all the chords for each major/minor scale, label each a tonic, predominant, dominate, and then try to find all the correlations. I'd like to see if there is a better way to do it. If not, I'll probably to try write a little program that'd list all the information. I feel it'd be about the same amount of work.
By learning the cycle of fourths/fifths, you'll have all the notes/chords in order. Draw a clock face, and by putting C at 12 o'clock, count round in fourths, so the next few are :- F, Bb, Eb, Ab. At the 6 o'clock point, the Gb can become F#, and eventually you'll complete full circle/cycle back at C. Any dominant is one step anti-clockwise; any pre-dominant is two steps.
Yes, that's a standard method of moving to a new tonal centre. Other good ways include using a pivot note - go directly from C to Eb by considering each as a good harmony for the note G. Or the Nike modulation - 'just do it'. An unprepared jump into a new key can be very effective. The 'dominant of the dominant' technique is, as @Tim says, described by the cycle of 5ths. If you're going to learn anything by rote, make it F,C,G,D,A,E,B (the order that sharps appear in a key signature) and B,E,A,D,G,C,F (the order flats appear). You'll notice it's the same list, backwards. There are mnemonics. Father Christmas Goes Down All Escalators Backwards. BEADs Give Catholics Faith. Also note that the pre-dominant will be one tone up from the new tonic. D7. G7, C, D is one tone above C. (I wonder if you could take the short path from D to C, rather than the long way round vis G? D, Db, C. Hint: the answer's yes. You've just discovered the b5 substitution! Enjoy!)
Modulation is defined as easing into a new key by various harmonic and contrpunctal means.
There is, of course, the tonal jump, which is not a modulation.
Then you have many ways of modulating into another key.
By definition, or nature, a new key segment has to establish a tonal center, which is what the purpose of the tritone (augmented 4th/diminished 5th) is all about.
You can in basic terms, divide the modulations into three categories: diatonic, chromatic, and enharmonic.
The key concept is the leading note, which always creates some tritone relationship with other notes. Depending on the circumestances, you could stretch the modulating progression, or keep it short and sweet.