American Crafts Cardstock for diecutting on the Silhouette--it is awesome on any electronic diecutting machine and does not burn up blades like Bazzill does. I also like Color Mates but can't find it anywhere.
Bazzill Coconut Swirl (I think that is the name) and Neenah Cardstock from the office supply store are my faves for Copic.
I like Recollections, especially for card bases, and I buy it in packs at M's,,,,it's quite sturdy, and even seems to be holding up to copics,,,,I'll probably get better for stamping and using copics on, but for everyday, love the stuff!!!!!! I wish it came in more colors though,,,,,,I buy Paper Co cs for other colors,,,,not as good, but great for normal scrapping,,,,,
I only use CTMH. It is a nice medium weight, white core. I use it for both cards and scrapbook pages. It cuts well in the cricut too.
I do a ton of stamping, but don't use copics yet.
Love CTMH for scrapbooking and cards...it works ok with copics (in my limited experience) but is not as smooth as I would like and the ink "bleeds" just a little if you are not careful near the edge of your stamp. (But that could just be a beginner mistake on my part.)