I have used silk thread and love it. Many needle turn appliquers have recommended using silk thread. It glides through the fabric VERY smoothly. You have to knot it onto your needle or it will come unthreaded every time you take a stitch.
Many other needle turn appliquers have said to use a good sewing machine thread: Mettler, Guterman, Aurifil, Masterpiece. These people say that the silk thread isn't as strong, and also that since you're using cotton fabric you should use cotton thread so that it all launders the same. So, I have also used Masterpiece Thread on pre-wound bobbins from Superior Threads: a "Frosted Donut" which is a round plastic bobbin holder filled with 36 colors of thread on pre-wound bobbins. This was recommended to me by several needle turn appliquers because it's so easily portable to carry with you when you're appliqueing on the go. Compared to silk thread, this thread also works well and it IS much more easily portable, but, it definitely is NOT as smooth going through the fabric. If your LQS does not carry "Frosted Donuts" you can see them and buy them online at www.superiorthreads.com.
As I've said above, the needle turn applique "experts" seem to be very divided (and also VERY opinionated) about which thread is the best to use. I'm not convinced that it matters that much.
So...I use both silk thread and thread from my "Frosted Donut". I actually prefer the silk thread because of how smoothly it glides through the fabric, so I use the silk thread at home and I use the "Frosted Donut" away from home because it is so easily portable. The fact that I use both (possibly even mixing the two on the same project!) probably horrifies the needle turn applique "experts", but that's what I've found works best for me.
As for the straw needles, the experts are again very divided and very opinionated about what needles to use. Some say to ONLY use straw needles, some say to use applique needles. Straw needles are very long and very easily bendable, much more so than applique needles. I would say try both and see which ones you like better and work for you better. The only things that matter are that it works for you, you enjoy it, and that your applique looks the way you want it to.
Hope this helps!!
Becky