I've found sourdough very easy to use, and it's completely free of most allergens other than grain and wild yeasts. I just found instructions online somewhere, probably here. This site has some photos of what starter looks like as it grows.
It's supposedly a myth that you are catching yeasts from the air. The organisms are actually in the flour itself, so the fresher and better quality your flour is the better it will work. Stoneground whole rye flour is supposed to be the best; you can always convert it to whole wheat or whatever you like once you get the starter established, just by feeding it with a different grain.
You'll want to cover it loosely to keep dust and bugs out, but it does need to be able to breathe a bit. I use a crockery with a loose-fitting lid or just cover a glass bowl or measuring cup loosely with plastic wrap. Make sure you have it in a large enough container that it can double (or, at the beginning, maybe even quadruple) without overflowing.
Basically I mixed equal parts (by weight, not by volume) whole stoneground rye flour with filtered water and kept that at warm room temp (75-85 F or 24-29 C) for several days. I fed it every 12 hours or so until it was well-established.
It's easiest to measure by weight if you have a scale, but if not use about 1/4 cup of water and 3/8 cup of flour to start out with. Water weighs more than flour does, so you'll have about 1 and 1/2 parts flour to 1 part water by volume.
You can start out with a small amount and double the volume each time, but eventually you want to start taking out half each time you feed it, or it will take over the house. I just added another 1/4 cup water and 3/8 cup flour the first time I fed it, and then I'd take out half and add 1/4 cup water and 3/8 cup flour each time I fed it.
Now I try to keep about 2 cups of starter at any given time. Most recipes start out with a cup of starter, so you're taking out half each time you use it if you have two cups.
Mine started out with lots of bubbles but a very bad smell. That's from the wrong kind of bacteria growing in it, but don't give up on it. As the correct yeasts proliferate, they create an acid that kills the other bacteria. It may seem to have died for a day or two when this change in dominant bacteria is taking place, but it probably hasn't.
You just keep feeding it (but don't use it yet) until it smells more yeasty. That should happen in about 3 days . . . if it hasn't happened by a week or so I'd probably throw it out and start over. If it grows mold, you'll need to start over also.
It's ready to use when it smells right and it doubles in size within 12 hours. Mine usually doubles in 4-6 hours and then collapses by 12 hours, so you need to kind of watch it to see what yours does.
Now I just feed it once a day, or you can feed it and put it in the refrigerator and then it keeps about a week between feedings.
Each time you feed the starter, allow it to get to room temperature. It will probably start to bubble. I keep about two cups of starter going at a time. When I feed it, I usually take out one cup of starter and add about a cup or 200 grams each of water and whole wheat flour to that cup to make the sponge for my loaf of bread.
To the remaining cup of starter, I add 100 grams each of flour and water (about 1/2 cup of water and 3/4 cup of flour, or a little less), which will bring the amount back up to about 2 cups. Let it sit at room temperature until it starts to bubble. You can then put it back in the refrigerator if desired.
If you keep the starter at room temperature it should be fed at least once a day. If you continue feeding it every 12 hours it will have a milder flavor, but it generates a lot of starter to use up. Most people probably aren't baking twice a day every day. :)
I've let my starter go 2 days or more at room temperature without killing it, but it seems to stay healthiest if fed every day. If at any time it grows mold, it should be thrown away.