Sorry for the crude drawing, I hope it shows the design well enough.
First of all, why not only use the bungholes on the bottom? Because air has to be able to escape. You can see that in this design, the top are connected with a pipe which connects to an air vent pipe so air can escape and enter as the water level changes. Remember to put mosquito screen on the air vent.
The pipe coming in should be as big as possible, because a lot of rain may arrive at once. The overflow pipe should be as big as the pipe coming in, so water won't back up into the downspout. This is a sealed barrel system, so you can use an automatic diverter which can deal with the tap pipe being full of water, but the air vent pipe opening must be higher than the diverter is.
The air vent is connected to both the top and bottom pipes so air can easily escape as the pipes fill with water. The separate vent should reduce gurgling and speed the flow of the water. The same principles are used in household drain systems.
The large vertical pipe on the left carries the water down from the intake, so it should be at least as large as the intake pipe. The horizontal pipe at the bottom should also be that large, so all the water coming in can easily reach the bungholes and flow in to the bottom of the barrels.
This is where the first bottleneck can be expected. Your downspout is probably 2x3 inches or 3x4 inches, so a 3 or 4 inch diameter pipe can handle it all; if your diverter has a smaller pipe then that's the largest size that you need (you might consider a larger pipe across the bottom in case you might be cross-feeding from a second downspout). But your bunghole has a two inch diameter -- or if you're using 3/4 inch pipe threaded into the bungs then you only have the 0.75 inch diameter to fill the barrel through. Obviously you want the horizontal pipe to be larger than the openings into the bungs, otherwise all the water will try to go in the first barrel and the other taps will be starved.
The cross-section of a two-inch pipe is 3.14 square inches (if you're doing the math, remember the radius of a 2-inch diameter pipe is 1 inch). The cross-section of a 3/4 inch pipe is 0.44 square inches. If you have a 2-inch bottom pipe then it can feed 7 or 8 pipes with a 3/4 inch diameter.
If you're using two-inch feeds into the bungs, but only have a 1.5 inch diverter pipe coming in, then you won't have enough water coming in to fill a single 2-inch opening, so there's not much of a problem in that case.
If you're paying attention, you might have noticed that I haven't explained why I'm showing the water-filled vertical pipe on the left having a connection to the top horizontal pipe. That's because I'm using that to cheat. If the bottom bungs can't let water into the barrels as fast as it is coming in, then the water will flow across and also try to fill the barrels through the top bungholes. There will be a steady flow of air coming out, but some water will probably be able to flow in through the top. There is a risk of bubbling or gurgling noise from this, but I don't know how it compares to the sound of the rain.
Also, even if you're feeding two-inch diameter taps into the bottom bungholes, a 3/4 inch pipe is sufficient for air to flow out of the top. This is useful because barrels tend to have bungs with different threads -- the bunghole with the NPT thread ("fine thread") can fit two-inch diameter threaded pipe. But the other bunghole has buttress thread ("coarse thread") and it's easier and cheaper to get buttress bungs with a 3/4 inch NPT knockout, then use that for 3/4 inch air venting. If you tap the barrels while they're upright, it will be easier to keep track after that which side should be up.
EDIT: Notice that this design is based on using rigid plumbing. Using hoses requires extra care to not trap too much air or water in curved hose sections.
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