I would totally suggest something small to start off with, because making even a twin sized quilt as a first project can get overwhelming. Were you planning on hand quilting or machine quilting (not the piecing but the actual sewing of the top, batt, and back together). Smaller projects are also much easier to manuver in the machine when you're quilting.
It can also be fun to piece together a whole bunch of different squares (for a sampler type quilt) and then sew them together to make your first quilt top. If you go a little at a time it's not as intimidating
a regular quarter of a yard is a bit long and skinny and not ideal to work with when it comes to quilting. To get a fat quarter you take a yard of fabric and cut it into quarters. One quarter is a fat quarter.
here's a ganked pic from about.com that explains it really well:
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Amie... we're finally moved in!!! YAY!!!
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