How many dedicated lines?

You might think about this from the standpoint of how many components in your system draw large amounts of current. For example, if you have two power amps that both draw significant current, you might want to have a dedicated line for each of them. Your TV also draws a fair amount of power, so maybe a dedicated line makes sense there also. If you have 1-2 large subs, they might also warrant a dedicated line they could share. The final remaining component that might also merit a dedicated line is your preamp/processor. Although the pre/pro draws a relatively small amount of power, you may want to insure that it has its own source of clean power, given that it feeds the rest of the downstream system. Other components that draw relatively small amounts of power, such as a DVD player, VCR, etc., could be lumped together onto one line. Counting up, then:
1 line for each power amp
1 line for the TV
1 line for the subs
1 line for the pre/pro
1-2 lines for everything else

I have to say that 6 lines may overkill, unless you live in area with very bad electrical power. However, it should ensure that you have lots of available, clean power to the components that need it most.

Try one heavy line for high current analog. One smaller line for all other analog. One line for digital + TV. Temporarily install them lying across the floor, then listen & experiment. Try adding more lines if you really feel the need & see if you hear differences. Experiment with directionality; some report that one way betters the other. Be sure to allow adequate time for conductor breakin.
I ran my own single dedicated line: three strands - the usual black, white & green (with some advice from Mike VansEvers) using #10 solid copper THHN. Line them up straight, & tape the ends all together. Put that combo into your power drill chuck & fasten the other loose ends tight in a vice. Now spin the whole bundle slowly until tight like a spring. It will unwind a lot when you power off the drill. Exchange ends & finish the twist. Run this bundle from the fusebox to a dedicated outlet. BTW: a 20A ceramic fuse sounded much warmer & smoother than the glass fuse. Regarding directionality: you just look at the printing on the wire's insulation & ensure that all 3 conductors are aligned identically. One direction may sound better than the opposite direction - you can try running the bundle either way (installing it temporarily of course). Re: the twisted conductors - this relates to something about the magnetic flux fields canceling each other. You'll find that some upgrade AC cords are made this way too.
Of course you must run this twisted wire bundle in a conduit (or in 3/8" Greenfield if code permits - much easier to work with). Regarding surge protection & filtering: I have a big Joslyn gas-discharge primary arrestor, across the whole house' primary, in the basement. Then a G.E. MOV (metal oxide varistor) in parallel across my Wattgate 381 outlet (mounted in a box) upstairs. MOV's are not supposed to degrade the sound; Chang Lightspeed uses them internally for transient protection. I also use two Chang Lightspeeds (a 3200 and a 9900 Amp) in addition to some pretty seriously expensive upgrade AC cords. The dedicated line sounded even better with the Chang's than it did standalone.
I'm gleefully satisfied with the results so far. Dramatic improvements over the house wiring! I polished the ceramic fuse clean & bright with crocus cloth, then applied Kontact. Same with the #10 solid conductors' ends, just like I do with all my AC cables. If you use anything larger than #10 it becomes a bear to work with. #10 is stiff too, but at least it's still workable. Also consider isolated grounding.

The fewer the number of lines, the less risk you run producing ground differentials and loops which may induce hum into your system. My two channel audio system performs best on a single circuit.

I like the idea of dedicated lines with dedicated neutrals and isolated ground if possible.Use IG recptacles. If you don't have enough breaker space add a sub panel close to the gear. SDcambell's lay out sounded like it would be a good place to start. Go with 20 amp circuits. Up grading to a # 10 wire is fine though I personally find that a little extreme on runs less then 150'. But what do I know I'm just an electrician not an engineer.

It really depends on the draw of the units....
In an all audio system, I use two 30 amp dedicated lines, one for each big Krell amp. And, I then have one 20 amp dedicated line for the front end. This, in my opinion, is just right, or maybe a little overkill.
Even one dedicated line is the most cost effective upgrade I ever made.