I don't know what it is like for the Civic, but you may want to have someone look up proper ORDER of bleeding the brakes. Rather safe than sorry. My Accord is somewhat of a strange system because it is split in two seperate hyrdraulic lines apparently at the proportioning valve or combo valve, or at least that's the reason I come up with,
I haven't actually looked at it to see if that's true or not.

The way the accord is setup is to start the rear passenger, then the front driver, then the rear driver, then the front passenger. My accord also has ABS and this is the only vehicle I found that I had to bleed that too, so if you're has that, it might also be a good idea to flush that out (just as preventive maintenance), ABS modulators are not cheap and flushing it out with clean fluid helps the different valves and seals maintain their efficiency.

Oh and definitly only three pumps is needed before holding, not 20, you could always place a small piece of wood in between the pedal and floor board so you don't hit a rusty spot in the MC and tear a seal.