An Accuquilt Go! would help you with cutting accurately.
For accurate piecing, I swear by having a physical "stop" on the bed of my machine. My current favorite is this adhesive guide:
http://www.amazon.com/Dritz-Quilting...dp/B0085L2Y2I/ . Read the reviews to see how to position it (or I can post directions here if you are not sure). When sewing, all I have to do is make sure that the fabrics are butted up against this physical stop.
What thread are you using? I found that my seams became more accurate when I switched to Aurifil 50wt 2-ply for both top and bobbin. It is a very smooth, thin thread.
The above suggestions are my most magical.
Ironing is a problem for some quilters, as ironing can either help with accuracy or distort your block. Do you use steam?
Personally, I do not like to participate in swaps because of the accuracy issue. It would take too much time for me to ensure the kind of accuracy I would like to deliver to others. Plus, I have seen the huge variations in block size that can come from a group of quilters! Instead, what I look for are patterns that are easy and somewhat forgiving. For example, if I want to make a star quilt, I look for a pattern that "floats" the stars -- never a cut-off star point that way!
Where are you taking your classes? It may be that this is not a good venue for you. You may need to stretch your quilting wings a bit.
p.s. I should mention that my sister has great difficulty sewing straight seams, even with a physical stop, as she is disabled in several ways. However, she has still put together wonderful quilt tops! When I get them, I fix any big problems -- such as gaps in a seam or 1/16" seam allowance, or even holes (I patch with a fusible) -- before quilting. When quilting her tops on my frame, I use Glide thread (never gives me problems) and quilt very closely together (usually loop-de-loops). The close quilting takes care of any "bubbles" and other inconsistencies in the top. They look terrific when done!
Don't lose the forest by looking too closely at a tree! Accuracy isn't everything when it comes to quilting.
Edit: Regarding the Dritz adhesive guides, I advise *NOT* using their red ruler to position the guide (comes in a different package). I checked their red ruler, and it positions for a true 1/4", not the scant 1/4" we need for quilting. It must have been designed by a non-quilter.