Dawn dish soap works well for wet felting. Any dish soap you have on hand will probably work.
Craft felt sheets have already been felted and if you can tear them apart, they have not been felted very well and probably have only a minimum of animal fiber. Craft felt sheets are made of many different things including recycled plastic bottles. Craft felt sheets that have a high wool content generally are thicker, denser and should not be easy to pull apart and are not what you want.
Wet felting is done with water and dish soap (or olive soap) rather than using felting needles.
To make acorns you would bunch a small quantity of fiber in your palms, wet it with hot water and a bit of soap and roll it in your hands until it is forming a nice ball. Then you would begin manipulating the ball into an acorn shape. Once you have the shape you want, you would rinse the fiber to be clean of soap, reshape it and let it dry.
It takes a bit of experience to learn to judge how much fiber you need, but you should be able to learn that quickly.
Did you buy craft felt cloth?? (guessing from your comment that you got the wrong kind)
Wet and needle felting is done with loose fibers (and there are so many kinds).
Animals are more likely to be made with needle felting.
I think you will probably find a larger selection of kits looking online. They are a good way to try out needle and/or wet felting as you minimize the investment into materials until you know whether or not you like it.
I am not familiar with AC Moore. Last time I was at a Michaels they had nothing related to felting (except a few books). JoAnn's carries Clover's felting supplies, but I don't recall there being kits (which could be that I wasn't looking for them).
For wet felting, you just need fiber, bubblewrap and dish soap. There are a lot of types of fiber and they can have a different feel, felt a bit differently etc.
For Nuno felting you'd need the same things as for wet felting plus some light weight natural fabric such as silk or cotton.
For needle felting you need a good work surface, needles and fiber. For just starting out, using a dense foam pad makes the most sense, but isn't the best work surface. I suspect that craft stores that sell fiber for felting are probably selling Corriedale. It's a good fiber for a beginning needle felted. Needles - there are a lot of different needles available, but for just starting out buying 5 of one size (#38star or a medium gauge needle) will be enough.
If you discover that you really like felting, there are better places to buy supplies online than brick and mortar craft stores. If you would like to know my favorite places to get supplies, send me a private message.
Acrylic fiber doesn't "felt" because it lacks the scales present on natural fibers such as wool and alpaca. You can, as you have done, share enough fibers back and forth between the layers of the yarn to cause them to stay together.