1. Starch your fabric before cutting. I use spray starch; Best Press works too. Starching helps prevent your fabric pieces from becoming distorted as you tug during sewing. I say "helps" because you still have to be careful while you press not to distort your pieces or they won't fit together.
2. Choose a large, gentle curve for you first project. My first quilt with curved piecing was Aquarium, using the Drunkard's Path block in a large (8") size. The complexity comes from strip sets that are pieced prior to cutting. This pattern is in the book Strips 'n Curves by Louisa L. Smith.
3. Sew with the concave piece (curve going in) on top. Ignore any directions that say it doesn't matter; it does matter. You will have an easier time with the concave piece on top, avoiding puckers and misaligned pieces. After you have lots of experience, you can try it the other way. You will then know how it is supposed to work and will be able to tell when things are going right or wrong.
More large curves; from Winners Bouquet by Atkinson Design
4. Slow down, take your time. This is especially important while you are machine sewing. You want to sew at a speed where you are in control of the stitches. This will be slower than the speed at which you chain piece straight edges but faster than you can hand sew.
5. Use an awl (I use the business end of a seam ripper) to hold pieces together and guide them close to the needle. It is tempting to use your fingers to control and manipulate the fabric but you can't let your fingers get too close to the needle or this is what happens.
6. Don't be afraid to sew from the center of the piece out to each end. Yes, this means you will have to sew twice to finish the entire length of the seam, but it takes less time than unsewing and sewing again. If it gives you better, more accurate control it will be worth it in the long run.
Double Wedding Ring
work in progress
8. It's a hobby, it is supposed to be fun. Give curved piecing a try, but if after some focused effort, it is more frustration than fun, move on to something else. There are lots of quilt patterns (like curved log cabin) that give the illusion of curves while using all straight piecing. Try one of those instead.