A rainbow beacon is essentially having pistons push stained glass over the light of a normal beacon at a specific rate. The timings are set with a gigantic pulser with six one-second long pulses. The pistons themselves overlap with the timings a bit so that the colors can blend.
The trickiest part of this build is placing the circuits where they need to go without interfering with each other. Using glowstone ladders is a good idea. Using different colors for different pistons is highly recommended.
To start the pulser, use a stone button, since it's on for exactly one second. One second intervals are important. I used redstone repeaters on the fourth setting to keep each piston on for 1.4 seconds so the colors had a chance to overlap. Glowstone ladders come in handy here since they allow you to transmit a redstone signal straight up.
It's also important to know the parts of this build. One mistake in this can take a little while to be fixed, especially if you know nothing of this build.
It's best to build this underground in a large area. Also make sure that the beacon itself has open access to the sky. When done right, you will see a beautiful rainbow effect. It looks especially good at night time!