Linen Weights for Cotehardies

Generally, I'd start with a 5 oz linen, lined at least in the torso with at least a 3.5 oz linen. If you're not going to line it, you might go a little heavier.

As far as thread, you didn't specify whether you were handsewing or machine sewing. For machine, I try and stick to Gutterman or Mettler, rather than Coats and Clark or less expensive threads, as they tend to make my sewing machine happier (fewer tension problems, etc). Stick to the standard polycotton, plain cotton isn't strong enough and silk will rip your fabric, eventually (it's very strong, and ideally your fabric should be stronger than the thread as otherwise the fabric will tear at weak points rather than the stitches breaking). For handsewing, you can use silk, fine linen, or the same thread you'd use in your machine.