There is a lot of great advice already given. However, this is what I would say if you really are looking just for bare essentials:
You might want to consider starting with partial boil extract kits for a few reasons. It is the easiest way for a beginner to get into the hobby, and many or most homebrewers start out that way. There is less that can go wrong than all grain so it will give you a chance to figure out proper cleaning, sanitizing, and fermentation. You will be able to learn the process and all the equipment you use for extract can still be used if you decide to move to partial mash or all grain. Things like a wort chiller are great to have but you can do just fine with an ice bath to cool your partial boils to start. A large kettle and propane burner are necessary for full boils (although some stoves can handle the full boil, it is extremely slow) but a five gallon kettle on a stove will work fine for partial boils. If you do decide to move to all grain, that five gallon kettle will still be useful for heating sparge water. There is also no reason you can't start out with a larger 10 gallon kettle and boil what you can on your stove until you want to move to full boils. You can get a good 10 gallon aluminum kettle for $40-50 that would work just fine.
I would suggest a starter equipment kit like
this one
Then add a kettle, a thermometer, a large spoon, an auto-siphon (not absolutely required but it will make transferring much easier), some way to control fermentation temperature (swamp cooler out of large storage container is cheapest option), starsan or some other no rinse sanitizer, PBW or another cleaner, a scale, and a brewing book like How to Brew by John Palmer
Then buy an recipe kit and be ignore the instructions (use How to Brew or some of the guides on this site instead)
Again, this is just what I would think of as the bare essentials, and I may be forgetting some things. There are plenty of things you can add on that make the whole process easier and faster (many already mentioned by others) so what you start with really depends on things like your current budget and if you think you are likely to stick with it for a while.