I got rid of stilt grass in my lawn by applying crab grass preventer aka 'pre-emergent' fertilizer that is applied w/ a spreader. The key is to prevent it from germinating, rather than trying to kill it once it is actively growing. I live in northern NJ & apply the crab grass preventer twice, once early in the season, usually early to mid March, then again 6 to 8 weeks later, which typically sometime in May.
As far as the other weed in the photo, I'm not sure what it is, but the same principal will resolve the issue, apply a pre-emergent & it will prevent it from growing in the first place.... however, if you forgot to apply, or miss an area w/ the pre-emergent, etc, it's easier to deal w/ than the stilt grass. The trouble w/ the stilt grass, is that once stilt grass is actively growing, it's difficult to kill w/out also killing the lawn surrounding the stilt grass, since it's a 'grassy' weed, & the only thing that kills it is 'weed & grass' killer.. so again, the only way to deal w/ stilt grass is to prevent it from growing in the first place.
Back to the other weed in the photo, first priority would be to prevent it from growing in the first place, as mentioned above.... but because it's not a 'grassy' weed, there are products available to kill it once it's actively growing, what you need is 'broad leaf weed killer' it's available in both granulized fertilizer form, which gets applied w/ a spreader to the entire lawn, and is also available is a spray, which can be used to spot treat affected areas.
Just remember it's a process, & isn't going to happen overnight... you need to develop a plan of what you will apply to the lawn throughout the season over X amount of months period... & after 1 to 2 years, you will see improvement & ultimately become weed-free... but it does not happen overnight, it takes time.
And there you have it in a nut shell, the knowledge that I gained after about 5 years of trial & error to get my lawn weed-free, after TruGreen could not do it.
