I have a hybrid brew-in-a-bag (BIAB) setup where I use a cooler fro my mash tun, but instead of a false bottom, all my grains go in a fine-mesh nylon bag. Makes clean up pretty easy! Most BIAB setups have the mashing bag go directly in the brew kettle. As with many aspects of brewing, there is no single way to get the desired result, though some techniques are more common than others. What you need to mash is a container which will allow you to mix ~5-10 lbs of grain with ~1-4 gallons of hot water and hold the temperature for 30-90 minutes, and then allow for some way to separate the spent grain from the sweet wort.