For making fabric bags, in addition to doubling the fabric, you could also use a piece of "interfacing" to reinforce the fabric (buy it on the bolt from a fabric store... there are many weights and types... some can be ironed right onto the fabric, and some aren't "fusible" so are just used as a second layer then sewed only into the seams to hold in place).
You could also make the bag with a separate inner "lining" that wasn't connnected to the bag except at the top... I've made loads of tote bags using the technique Eleanor Burns put at the end of her first book, which you can get cheap if you're interested:
http://www.amazon.com/gp/offer-listing/0922705984/ref=sr_1_olp_1?ie=UTF8&s=home-garden&qid=1242929479&sr=8-1
And there are other ways of making liners in tote bags too. Liner material can even been water-resistant or water-proof (and some people are making them totally from thickened plastic bags, etc., too these days).
The most stress for fabric bags would be on the bottom, the seams, and places the handles are attached, so you could just make those areas stronger too (with additional stitching, e.g.). Or you could put a stiff "floor" in the bottom of a bag you'd given a wider bottom to (by turning the finished bag inside out, flattening the bottom area as if it's sitting, then stitching across the triangle corner of the fabric at each end of the bottom seam...here are a couple of pics of making that kind of "box" bottom from a regular bottom:
http://www.wikihow.com/Image:T4_935.JPG
http://www.craftstylish.com/assets/uploads/2008/06/Fig-11---Make-Mine-3D_lg.jpg )
And here are a bunch of lessons on making shopping bags and tote bags in general:
http://www.google.com/search?q=how+to+make+shopping+bags
http://www.google.com/search?q=how+to+make+tote+bags
HTH,
Diane B.