Okay, I'll say it before all the rest:

The only completely chemical-free way to get weeds out of the lawn is to pull them up by hand. Preferably within a day after a wallopin' heavy rainstorm. Or after the lawn has been sufficiently irrigated so as to have the soil wet at least 3" down - if you're going after dandelions, it's best to have soil wet about 5" down.

It really is not that big a deal - although how big depends on how many weeds to the square foot of lawn. Start at the farthest point and work your way backwards, so you aren't walking over the area just weeded. If you have reasonably good soil, your weight is not going to compact it much, even though it's wet, and getting up the deep roots is a wonderful aerator. If you can get most of the weeds in the early spring, there won't be too many the rest of the year.

Speaking from experience, the first time, the first year (after we bought a previously owned and very neglected house) it took me about 10 hours each day for over 3 days to clear mostly dandelions and wild violets from the front lawn, 100'x125'. They were really healthy d'lions with taproots that went down more than 18". There were literally more d'lions than grass. I threw grass seed over the area after I finished (not the proper way to do it, but by that time, I didn't really care!) which did pretty good at filling in the bare spots. I had to pull weeds for a couple hours every few days April and May because the baby weeds kept coming up, but by June there was a pretty good lawn. The second year I pulled for maybe an hour a week for those months, and since then I pull for an hour or two in early April and again in mid-May. That gets most of the weeds. The violets are far more difficult to eradicate since even a tiny piece of root will make a new plant - I'm still pulling them out of the lawn every spring but at least now I have to look for them hiding in the grass. So it IS possible (just very time-consuming) to get rid of weeds without resorting to poisons. In the years since I have tried several home-remedy-type applications of which boiling water or full-strength (5%) vinegar are more-than-less effective for spot-treatments. I'm experimenting with the use of fine corn-meal as a pre-emergent weed treatment, but I haven't been able to decide just how much it helps - mostly because I'm not sure just when is the most effective time to spread it.

There is a hand tool called a "weed-hound" that works OK on
dandelions. The soil must be damp for good results. Some
of the root, maybe 2 inches, will be removed, along with the
green parts of the plant. The tool is fast, you can do 3 or
4 plants in a minute.

Hand dig is right....buy a dandelion hand fork and you can get most of the root without disturbing the grass. Then rent an aerator, use an organic lawn fertilizer (Ringer Lawn Restore is great, there are others), and mow often so you are not removing more than 1/3 the height at any one time. Before you know it, your lawn will require less water, be so thick weeds don't establish, and the few that do show up can be easily hand dug....

And there are 'flamers'. Strange-looking gadget that has a long handle and uses a small propane tank for fuel. You literally flame the offending plant although many folks don't realize that you aren't supposed to burn the weed, just heat enough to bubble the juices. Very effective, especially if you can't get down on your knees to pull up offenders. Here is a link to the one I use.

Here is a link that might be useful: weed flamer

I have had much luck with household vinegar, boiling water and ammonia poured on weeds in weedy patches. You can also use baking soda. For the areas with weeds next to grass though, I hand pull.

Weeds by definition are the plants you don't want. Meldy mentioned the difficulty of removing violets; I say, why bother? They have pretty flowers in the spring. Likewise, I don't try to kill clover, which fixes nitrogen and used to be considered desireable in lawns. If you want a perfect, golf-course type lawn, you're better off using chemical weed-killers and lots of fertilizer. Why not go for a more natural lawn (but don't tolerate dandelions, plantain, or creeping charlie!).

Use mulch it works for me and mulch is cheap