I have just a regular sewing machine, and I was wondering if I can use the cotton machine quilting tread that are on cones for piecing. If I can, what could I use to hold the cone on?

I use the big comes on my regular machine all the time. I have a thread stand that I got from Superior Threads. I had a plastic one from Joann's, but for me it was too light weight. Superior's ar much sturdier.
I've also seen posts about using a mug to put the cone in, but I haven't rried that.

Oh definitely, I only buy cones as they're much more economical. Superior Threads stand is cast iron and can't be beat.

Sick the cone in a mug. :) Works great. There are also thread holders you can get, but I just use a mug.

By the best thread stand, a heavy can't be tipped over by the cat one. You'll never regret the purchase. Connecting Threads has a beautiful one and a nice basic one. Do not buy the plastic one, it will turn over while you are sewing once the cone gets half empty, no weight to hold it down.