For raw edge, I use either Misty Fuse or Pellon 805 (Wonder Under). I window it (see Pat Sloan's demo here:
https://www.youtube.com/watch?v=r_m84NtcyUk) regardless. Pellon is heavier, but seems not to fray quite as much. It does produce a noticeably stiffer end result, but sometimes I don't mind that.
I only use cotton thread for applique. For some reason, I always end up with a lot of issues with thread breaks when working with poly or mono threads for applique. I pick colors that match the applique piece & usually like doing a scant 1/8" (2mm or a little less) blanket stitch. It is dense enough to keep the edge fairly secure through multiple washings ... which is the most important aspect to me as I primarily make baby quilts. I've worked with both a Gutermann 50/3 & Aurifil 50/2 cotton thread. I must say that I do love, love, love Aurifil. It comes in so many colors & always produces a stunning result.
Honestly, I was so reluctant to have to change threads so often for my applique, but it really didn't take a ton of extra time & I was so happy I did it. I used the same threads for quilting each of my appliques as I did for the appliqueing & it turned out really nice.
Grey thread will blend the most of any color if you really don't want to change threads at all. Otherwise, I've found that
http://www.hawthornethreads.com/threads/aurifil usually has the best deals on the 200m Aurifil thread & they stock every one of the colors. One thing I will share is I would
not ever use the variegated or sandblasted colors for most appliques because they variations of the color/saturation will show up differently against the fabric & can make it look like you missed stitching certain sections. For some reason the LQS'es around here often suggest variegated threads as something that will "match" print fabrics, or multi-colored applique piecing; I haven't been successful with them for that purpose.