Because you are doing hand applique I think it will be important to choose fabric with a close weave. A fabric that has a looser weave is going to ravel more and will make needle turning difficult if not impossible (especially if your new at it). The looser the thread count the more difficult it will be to get good straight lines and make nice smooth curves.
You should be able to see and feel the difference between a tighter weave and a looser weave. Some extreme examples are a batik (tightly woven) and a homespun (loosely woven). Or to use your bed sheet example of feel and look - the difference between a 200 thread count and a 600 or 800 thread count.
I CAN feel and see the difference in some of the Joanne's Fat Quarters I recently purchased for a small wall hanging project. I would not ever consider using these for applique.
Batiks are typically a tighter thread weave, and batiks are dyed. Calico's and other fabrics are screen printed. The main difference in dye vs screen print is that dyed fabrics the fabric is either in a dye bath or liberally painted on and the color goes through the whole cloth.
Screen prints use a series of die plates (large rubber or silicon material - like a rubber stamp), there is one for each color used. The fabric moves through a series of these die plates and is imprinted with ink on the die plate. The color "sits" on the top of the fabric, it doesn't go through.
And then there are homespuns, mostly plaids, and these are made by weaving different color threads - so like a batik the color is all the way through. These may still be quality fabrics, using quality cotton - but the nature of the weave lends them to a limited number of uses - typically not applique, but they are used quite often in pieced quilts. Although come to think of it, I have felt some homespuns that are a nice tight weave that I would use in applique.
The quality of the fabric has more to do with the type of greige (raw material) used, and the process used to weave it.
Some fabric, a lot of the Asian prints come to mind, are a thicker greige/weave, a bit stiffer to the hand. I personally wouldn't want to needle turn one of these.
If I were going to take the time to hand applique a Baltimore Album I would use the best quality I could find.
I'd start by visiting a LQS - finger all the different type of fabric. Purchase what you think feels like good quality, nicely woven, soft fabric. Buy a fat quarter of one batik and one screen print. Then go to Joanne's and do the same thing. Take them home and make small tests blocks of each of them - just cut a simple applique of a heart (has curves, straight lines, and points) and hand applique it to muslin (or the other half of one of the fat quarters) and see how each of the fabrics behave. You'll know more by working with each of them than you will from any of us talking about it.
And I have to add ... new to quilting and wanting to do Balitmore Album. You go girl!!!