I couldn't see what faucet you were referencing - it said it wasn't available or something (hopefully I just cut and pasted wrong...). But generally speaking, when you add a below-counter filter system, you simply unhook your water line, connect the filter system in its place, and hook the cold supply line back in on the other end of the filter. That will get you filtered water all the time (on the cold side, at least).

If, like it sounded, you're adding a separate filtered faucet just for drinking water, you would do something very similar - add a tee valve between the wall plumbing and the cold water line to your normal faucet. The tee would branch to your filter system and your new filter faucet would connect to the other end of the filter, while un-filtered water would happily continue on through your standard faucet. Make sure the tee is a tee valve (can be shut off separately), because it makes replacing filter cartridges much, much easier!

When they said you need a vented faucet not made for reverse osmosis, this is because you need some sort of air gap, and if you get a filter faucet with an air gap built in, you don't need a separate air gap cluttering your countertop. An RO faucet typically has three lines where a standard filter faucet only has two.