My number one approach to weeds--and the one I strongly recommend--is preventing them. Look at the seedhead of just about any weed (such as dandelion), and you will see that every weed you get rid of before it blooms will prevent thousands of new weeds from popping up. And vice versa.

Prevention is easier than weeding or spraying
If I don't do anything else with weeds, I try to keep them from going to seed. Even when I'm too lazy to dig them up or pull them or spray them, I still try to pick off the flowers before they go to seed. It's a lot easier to get rid of a flower full of seeds than seeds that have blown all over your yard and become weeds.

You have to follow through and dispose of the flowers and seedheads where the seeds won't blow around, or you might as well leave them on the plants. Pulling, digging, or cutting a weed seems to make it go to seed faster.

If I can catch weeds just as they're sprouting, I scrape them off with a hula hoe. You can also use this on older weeds with shallow roots. This method isn't very practical with hard, uneven surfaces and rocks. It also keeps the soil disturbed, which makes it more likely weed seeds will sprout there again.

Herbicides

With mature plants with tap roots, my only choices are digging each one out or spraying them with Roundup. I'd rather not use Roundup, so I'm motivated to prevent as many weeds from sprouting as possible (see above). As fewer weeds return, the less Roundup I have to use. Of course, RoundUp kills every plant it touches, so it can't be used in lawns or flower beds. I know farmers use it on crops, but I wouldn't use it on anything I plan to eat.

I've had pretty good luck with Preen, an herbicide that prevents seeds from sprouting. I use that along pathways and other areas where I know I don't want any seeds to sprout. I can't use it in areas where I count on annuals such as flax or hollyhock to return from seed each year.

I've learned that "weed and feed" type lawn fertilizers harm trees and shrubs in lawns, so I've become selective in using them on our lawns. I sprinkle Scotts Turfbuilder Plus 2 on the broadleaf weeds and use plain fertilizer on the rest of the lawn.

The villains
One of my worst weeds is plain old lawn grass anywhere I don't want it. Digging it out has never been effective for me. There are herbicides that kill grass and not flowers, but I haven't had much experience with that yet.

Even worse than lawn grass is salt grass. It will grow through anything, and you can't kill it. And once salt grass gets into lawn grass, it's there to stay.

I have better luck with the "broad leaf" weeds. The ones I normally have are dandelion, mustard, tumbleweed (Kochia), thistle, prickly lettuce, sowthistle, salsify, bindweed, common mallow, purslane, and spurge. I also have the dreaded noxious weed tall whitetop, but I'm able to keep it well controlled.

Weeds of the West
By the Western Society of Weed Science
Just the facts, ma'am: photos and descriptions of weeds. Maybe I'm weird, but I like to know the correct name of every plant I have even when I'm killing it. This book can help you decide whether you need to kill a plant or not.