If its being piped away from low spot, a big pipe with no barrel may be better option(if budget doesn't allow both).
Your description of a rainforest has me thinking almost daily rain.
Therefore, despite not seeing any pictures and with lack of really knowing everything about your situation, I would think another possible, perhaps more useful, solution might be to either dig out the front yard a bit to create a pond, or simply add buried drain pipes to carry the excess rain water away to another less bothersome location on your property.
P.S. If you do go the rain barrel route, ---If you never completely empty the rain barrels, you could consider adding a gold fish or two to each barrel to control mosquitoes. This old time trick, that I learned from the Amish, also works well for animal water troughs. I keep fish in the water tanks year around with no ill effect on the fish or cattle. Around here gold fish only cost $.25 ea. at Pet Smart.
David Ulrich wrote:Berming--before water settles in one area--as is a technique within...permaculture for getting water "caught" to where it's desired...being (e.g. watering some trees, etc.) There's... drain pipe (in-ground), etc.? Also...? "Swamp"...plants: wetland's varieties...desirable, actually?
David Ulrich has the right idea to use "rain garden" techniques with plants that thrive and tend to help manage the water in specifically designed areas. They most often use native plants. it helps to excavate a depression and help things along by adding gravel and other aggregate to help accumulated water soak in. High water runoff areas such as impermeable surfaces such as roofs and patios can have "rain gardens" statically placed immediately down stream from the drain off areas. Rain Garden Network here is an example of a link with info, one of many.