If you're interested in a basic primitive brew you can do it without buying an expensive kit, try making apple cider.
All you need is a 2 or 3 liter plastic bottle of apple juice or pear juice ($3), a spare lid for after the brew has fermented, a packet of brewing yeast from the brewing supplies shop (around $1) & an airlock with grommet (around $1.50).
Drill a hole the right size for the airlock with grommet in the lid of the apple juice container, then fix the airlock in the lid so it's nice and snug. Remove some of the juice - leaving about 3 or 4 inches of air. Pour in half the sachet of brewing yeast, put on the cap with the airlock. You can store the remaining yeast in the fridge or freezer in an air-tight container or the packet sealed well. Place water in the airlock so that no air can move freely through the airlock & place the whole thing in a warm spot. Within 24 hours the brew should start fermenting (it will take longer in colder climates) & if there is a high sugar content the bubbling can get quite noisy & violent, with a boiled egg smell. Within a week (or longer if cold), it should stop & smell very alcoholic. Throughout this time make sure the airlock doesn't run dry. Go ahead and have a sniff & taste of your brew. You can then add a tbsp of sugar and replace the lid with a solid lid with no hole, then shake it several times or until the sugar dissolves, if you wish. The sugar added now is for the fizziness. If you prefer apple wine, don't add any sugar & you'll be able to enjoy your apple or pear wine immediately. Leave for a week, then drink. If you leave it for a few months or longer, the flavor improves.
So, you get 2 or 3 liters of apple cider for about $5.50.
Of course you can dilute the apple juice to half water - that way you could do two bottles at the same time. If you do this, you'll need to add about 1 cup of sugar to each bottle to make up for the sugar you lose when diluting the juice with water. The more sugar you add, the more alcohol it will contain. I prefer this method, otherwise the brew can be very strong.
Once you get the feel of this it's easier to move on to the larger kits for a hops & malt beer. Americans would easily be able to adapt the basic recipes on the kits to make sarsaparilla & ginger beer. The advantage with this method is that you don't have to worry about sterilizing anything - you should do fine with washing the only added components (the airlock and grommet) in warm soapy water or sterilizing it with the sterilizing powder you can buy from the brewing supply shop.
If you have a glut of home-grown, unsprayed apples or pears, you could make a wild & primitive scrumpy.
You clean the apples in plain water, then juice them with the skins still on. Pour the juice into a similar bottle, say a 3liter bottle with a lid set up as previously mentioned. Dilute the fresh juice with water. No need to add yeast as there is wild yeast on the skin of the apple. Follow the rest of instructions above & a washed airlock with grommet is okay because nothing else is sterilized. I've had success with this recipe using windfall fruit. It always tastes great & the fact it cost me nothing is a bonus. I usually make mine in the 26liter brew kits & then bottle them.
Once the juice has fermented, you can skip the secondary fermentation for fizziness if you prefer it as a wine. With this wild fermentation method, you wont always get the same results as the yeast varies according to location and season.
You can use this method for just about any fruit except citrus fruits. I think you'll have to add yeast to those. You can also use unusual fruit like tomato or herbs or vegetables like pumpkin or a combination like beetroot & hot peppers. If you use fruit in season that's cheap or home-grown surplus or even from roadside trees, you can relax more about the experimentation as you wont be spending lots of money. You can alter the flavor of the brew with spices or adding other fruits like plums or even grapes.
At the bottom of the bottle you will see a layer of yeast. You can leave this in the bottom if you pour out the cider slowly, then reuse it by making another batch in the same bottle. If you want, you could pour the yeasty dregs into some flour & salt to make a loaf of bread. Can you tell I really hate throwing anything useful out. If you do throw the yeast into the garden, be aware that snails and slugs love it. You can use this to your advantage by leaving a deep dish of the stuff out near the places they love & the next day you will find a whole lot of drowned slimy creatures. If you use an old trash can lid, it makes for an impressive platter for the Chickens, and ducks of course will also enjoy this treat.
If the brew is unsatisfactory, you can turn it into vinegar. All you do is remove the lid & pour in a teaspoon of vinegar including the mother from a bottle of raw cider vinegar that you can buy from the health food store. Then cover with a cheese cloth or piece of very loose-weave discarded t-shirt - use a rubber bad to attach the cloth. Then leave in a airy place and wait for it to turn into vinegar. Takes a couple of weeks, but can just sit there for a lot longer & the acidity depends on the alcohol content. It's safe to taste the brew at any stage during the process.
Once you use a container for vinegar, don't use it for making beer/cider/wine as it may not work out or the alcohol will spontaneously turn into acetic acid.
If you have hogs or chickens, they will gobble up discarded cider mixed with their feed. Oh, small amounts for the hens though, otherwise your they will be legs-up for the rest of the day. Apple cider vinegar is given to most farm animals as a kind of tonic.
Don't be scared because very little has been sanitized. Brewing is an ancient craft done successfully long before sanitizers. If you do get a dud batch, you can still give it to chickens as long as it doesn't smell really filthy. Apart from that, all I can say is that over the years there will be a few batches that wont work out - even when you do sanitize everything. That happens to everyone, so don't give up if that happens to you.
I have read accounts about people making beer in large trash bags in a prison, using fruit cups and fruit scraps and sugar and wild caught yeast. They had no airlock, so the bag was tied and occasionally opened to let the gases out.
As with all diy things, use common sense so you dont get hurt or hurt someone else. When in doubt ask someone who would know.
If you did have a go at this, I'd love to hear how it went.