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Nope, you steep the grain in the mash tun at between 62 and 68 degrees depending on what you're making and whether you're going to do a step mash or not (that's for later on in your journey). Basically, the cool box with a towel over it keeps everything at around the right temperature for an hour. The one I've got (which I don't use anymore as I have an all-in-one system) never let me down - only ever lost around one degree over an hour, which is good going.
You boil the wort (steeped grain water) that you get from the mash process in the kettle (big pot with tap) before cooling it and sticking it in the fermenter with the yeast.
You can do a kit beer without any more equipment than that (assuming you've still got the stuff you made the wine with), but if you want to make better beer, you'd need some additional stuff.
You can do extract or brew in a bag, which means you only need to get a kettle (a 30 litre stainless pot with a tap on it will do the trick), or you can go all grain, which means you need more stuff like a mash tun (converted coolbox is a good way to go if you're just starting out).