What is the minimum equipment to start brewing some good beer at home?
Transferring day (if you transfer)
Nice to haves
This list is mostly if you plan on doing extract brewing. You might need extra equipment for a full-grain setup. Also, this assumes you are using a kit that comes with necessary ingredients and a muslin bag.
Brew kits, which should supply much of the equipment, are available from various homebrew shops.
I would point you to the Basic Starter Kit from Northern Brewer:
Some items are probably not STRICTLY needed, but for $80 you definitely get all the gear you would need to get started.
Assuming you're beginning from concentrated malt or a kit, and can figure out how to boil water on your own,
The bare minimum:
You can make a decent beer with just the minimum. Add the rest as time and budget allow. Each time you go to the supplier, buy one more thing to make the job easier.
Don't underestimate the affect good temperature control will have on your beers.
While not strictly necessary I believe to make good beer you should invest in some kind of fridge/freezer with a temperature controller. Often this can be done fairly cheaply if you can get hold of an unwanted fridge and pick up a cheap temperature controller from eBay or similar.
Before moving from kit to all grain brewing adding consistent temperature control during fermentation was the single thing that made the biggest improvement to my beers.