For baby quilts, I always use 100% cotton batting. Polyester doesn't breathe so the baby can overheat (babies are not very good at regulating their own body temp) and in addition, some parents worry about the fact that polyester melts into molten plastic instead of burning up like natural fibers. It can mean the difference between 2nd and 3rd degree burns in case of a fire. Rayon and bamboo (a type of rayon) fibers tend to have lots of chemicals in them because of the processing involved in making those materials & the batting can off-gas into the baby's room. Wool can be too warm for some babies and some little ones are sensitive/allergic to wool, but it is breathable & is better at wicking away moisture. In addition, wool is generally much thinner and lighter weight than cotton. Silk batting typically is part polyester.
Those are the reasons why I choose 100% cotton batting for all my baby quilts, but obviously there are thousands of people around the world who wrap their babies in woolen bedding, including quilts, without any problems. If I weren't going with cotton for a baby quilt, my next choice would be either a wool/cotton blend or 100% washable wool. I would only use polyester if I were making a wall hanging for a nursery. The risk of overheating just isn't worth the cost savings to me.
As far as what brand, that just depends on what type of quilting you plan to do & how puffy you want the quilt. I am making my first hand-quilted crib quilt right now & am using Hobb's Organic Cotton batting. It is just beautiful -- no scrim, all cotton & a little puffy but not too much to where it's difficult to push a needle through. Quilter's Dream Select (or it's thinnest cotton batting, Request) are very thin battings that are easy to hand quilt, but do not provide a lot of definition in the quilting.
For machine quilting, I absolutely LOVE Warm & Plush from Wal-Mart. It is a beautiful clean, unbleached cotton that is needle punched (no scrim). It is the fluffiest cotton batt I've found on the market & it is heavy. I've also used Warm & Natural from Wal-Mart/JAF. It is also an unbleached cotton, but it is only 2/3 as thick as W&P mostly due to the use of a scrim (layer of synthetic web-like material that binds with the cotton to hold it together). The scrim & lower loft are the two main reasons I switched to W&P, but some people also say that they don't like that W&N batting has some cotton seeds or teeny tiny twigs from cotton plant that are left in the batting. Those should only be noticeable on your finished quilt when it is wet or if you have an especially thin white fabric in your quilt (new cotton fabrics shouldn't cause that problem).
For larger, adult quilts, I would not hesitate to use Hobb's 100% washable wool and would be especially likely to choose wool if I planned to overquilt (e.g., microstippling) my quilt. Wool quilts the most beautifully of any batting out there -- good stitch definition in a lightweight material & it is plenty warm for any time of the year. I personally like the comfort of heavy quilts to sleep with, so I will sometimes pull my rainbow quilt off the wall to use as a lap quilt because it has the nice heavy W&P cotton batting. But wool is great for sleeping. I have some wool blankets that are 1/8" thick and they are ideal. I never wake up sweating or chilled with them. They are light enough for summer & warm enough for subzero winters.
But all of that is just my own opinions. Most batting manufacturers are wonderful about sending out samples. I know for sure that both The Warm Company & Quilter's Dream have sent out samples upon request. The best way to know for sure is to make some quilt sandwiches in the different batts ... just keep in mind that with heavier batts like W&P that you will actually need to fit that denser batting through your harp, and you will have to work your neck and shoulder muscles all the more to move it around if you plan to Free Motion Quilt it.