Welcome all the Totally Groovy quilt a long folks!
I love all kinds of piecing, but it took me awhile to love piecing curves with traditional templates. With anything, it takes lots of practice and repetition to develop that love. Back in 2010 I wrote a tutorial for piecing curves the traditional way...with templates and pins called "Conquer Your Fear!" I pretty much piece all my traditional curves this way except I now eliminate the pins. Check out that tutorial for all the basics of traditional curved piecing.
If you're just beginning, use some pins. I pulled out Emily's templates from the pattern you're using and made a few just so I'd know what you're tackling. Since the Conquer Your Fear Tutorial was done with half circle templates, and this quilt requires quarter circles, I thought I'd share the steps again.
First, cut your inside and outside pieces and press each in half to make a registration mark and place them as shown below. Notice the folds are opposite.
Alyssa is having a little giveaway sponsored by Fresh Modern Fabric to celebrate this tutorial and keep you motivated.