One of the main reasons for thread breakage is a needle-thread mismatch. Start with a really, really big eyed needle, like a top-stitch needle, as someone said. With it, use a fairly fine-weight thread at least until you've found a "safe place" with little to no thread breakage.
And try polyester thread -- a hush fills the room ;-), although the idea that you shouldn't mix cotton and poly is a well-repeated myth ;-). Trilobal poly will be less nubby than cotton and thus less likely to catch in the quilt sandwich and break. Leah Day (the 365 quilt block lady) uses Isacord poly, but many of the poly threads will do. I even use Maxi-Lock swirls, in my longarm quilter. Although it's actually a bit nubby, it's fine-ness makes up for it. It is fun and holds up really well! But a trilobal poly will generally be smoother. The high-end pros like Karen McTavish use Glide trilobal poly.
You also want the needle to be nice and sharp. It has to make a clean hole through the quilt sandwich to prevent shredding the thread in the fabric sandwich. So make sure the needle is new and not damaged.
And patience helps. Watch some of the quilting videos online. Use them to help you develop a medium-speed rhythm.
And yes, another vote for setting your stitch length to 0.