Maria, I'm a quilter. And when I call myself a quilter, what I mean is that I sew quilt tops. I've never actually quilted one; I always hire that out. There are people everywhere who will quilt your quilt for you once the top is done. Your best bet is to ask at a local quilt shop, they will have a list of people with the long-arm machine necessary to do the job. Depending on how big the quilt is, the price can range from $50 to $300. Or so. It also depends on the kind of quilting you want done.
Now. About this t-shirt quilt. Yes, it can be done. And it can be done by a novice. But the thing is, with t-shirts, is that they are on knit fabric so they are intrinsically very hard to work with. My mom (also a quilter) made one for her sister and I believe that what she did was iron the graphic she wanted to save from each t-shirt onto some stablizing medium. There are several different kinds available.
This link discusses the various types of stablizers available if you want to embroider on t-shirts; your quilt is a similar concept.
About different kinds of stabilizers. You'll want to find a way to keep the knit fabric from stretching as you sew the t-shirts into "blocks." I'd recommend using some kind of sashing between the blocks--a thin piece of coordinating fabric to sew your t-shirts to so that each is framed by it.
It's definitely not the easiest project to start with but it's not impossible. Good Luck!!
A t-shirt quilt is always so very very special because all the t-shirts give it special meaning to the recipient.