also check your stitch length ---- use a longer stitch, as they tend to get shorter with thick layers.
If you use safety pins as suggested that should help. I use warm & natural, but NOT bamboo variety -- it tends to be less stable and move around.

I recently had a twin quilt basted by my quilt shop. They charged .015 per sq inch. I did the math wrong (normally I'm good at math). I figured it was going to cost around $35. Imagine my surprise when it cost $95.

But I really like it. I quilted stabilizing lines around the 10 inch squares and did stitch n the ditch around the borders. I then removed the basting (very easy). I had no problems with the fabric on the back getting puckers. I'm now doing the quilting in each square.

My my cousin owns a long arm and has offered to let me use it. I am thinking about basting my own. But it was really great having it done.

I'm not sure what the sole of the walking foot is.
There were several things people asked me to clarify. My current machine is a Singer Scholastic, nothing fancy at all. It does allow me to drop the feed dogs, I tried FMQ with little success. I know that takes time to master, I also think that getting a slippery surface for the sewing machine bed will help because my biggest problem was that all of a sudden the quilt would slip and the stitches would become long.
I have tried a walking foot, but I get more puckering than when I use my zig zag foot.
I do have the quilt supported well, though the bed of my machine is not even with the table top, I quilt on the kitchen table.
I have not started my quilting by making 4 quadrangles. That is something I will definitely do! If I'm channel stitching the quilt would I just do one half and then the other?
I have a hunch that my machine, while fine for making clothes and other projects, isn't suited well to quilting.
Anything else I left out that would help you all to diagnose my problem? Thanks!

The walking foot is not meant to be used with free-motion quilting, only for straight sewing to help the top fabric feed evenly with the bottom fabric, so take that foot off as most are bulky and will restrict movement of the quilt. Your needle position should be set at zero with the feed dogs down. Try using a normal foot, or an open-toe one. If you still have problems you may have to get a 'hopping' foot, which has a spring, so the foot bounces as it stitches. Hope this helps some.

Do you mean you are using a zig zag foot when you drop the feed dogs? If so, this may be your problem. You need to use a hopping foot, darning foot, or floating foot to FMQ. A zig zag foot requires the feed dogs to be up and does not allow for FMQ.

Edit: Once you have the correct foot, you can experiment with dropping the feed dogs or leaving them up. On some machines it is easier to control the quilt with the feed dogs up. I FMQ on my machine with the feed dogs up, but I am using a hopping foot (darning foot).

I use poly batting all the time without problems shifting.....the key is basting well (glue lines about 3-4" apart)....I am more inclined to think that you need to reduce the pressure on the presser foot and to make sure you have the right foot for the right job....walking foot to do straight line quilting with the feed dogs up.....hopping foot/darning foot to do fmq with feed dogs either up or down (personal preference) with stitch length set to 0....on my machine, if I want to do fmq, I also have to set my thread tension tighter (8-9) to get a good stitch.

I'm sorry I was unclear about the different feet. I am familiar with which foot to use, walking and FMQ hopping one. I think the presser foot pressure may be an issue, I have grandkids and it may have gotten moved from the factory setting.
I also think that not starting in the center of the quilt and quilting in quadrangles might be causing puckering. Do you all quilt in quarter sections?
Also, is it important to have precisely the right needle? When I'm stitching through only two pieces of fabric the needle seems like it's not going through the layers of fabric without making an effort with a 90/14 that the guy who owns a sewing machine/vacuum shop said to use. That seems way too big to me. I had been using a 75/11 quilting needle for quilting and piecing. Can a needle cause a problem with the feeding of the quilt through the machine if it's not piercing the fabric easily?
And haven't been fussy about which thread I use either. I've simply been using Dual Duty Polyester, but my quilting book says to use cotton. How important is it to have a specific thread?
Thanks for the help!

I would suggest some instruction such as a Craftsy class. They are inexpensive, frequently $19.99 (like they are today). I especially enjoyed the class by Ann Peterson. If you go to the website, you can get little preview blurbs and lesson descriptions of the class. Some classes are quite basic and others are much more advanced. It is so hard to evaluate the difficulties someone is having by verbal description. It is easier to see someone else (who is skilled) demonstrate the skill and allow the viewer to consider a comparison of techniques and change styles.