Personally, my initial approach for emerging weeds in the lawn is to hand-pull.
(There's actually a step before this, though -- I use pre-emergent to give coverage about 8 months a year via 3 different applications of prodiamine at a 3-month rate, with some overlap in two of the treatments.)
Then, for anything which gets past the pre-emergent, my first line of attack is to hand-pull.
If there's too much of something to make hand-pulling feasible, I prefer to use "targeted" herbicides, rather than a 3-way mix.
I use 2,4-D on anything that looks like it belongs in a salad or that I can't identify.
I use triclopyr on anything that I know (from experience) won't be killed by the 2,4-D. For me, this is primarily wild violet and clover.
I use quinclorac (Drive XLR8) on infestations of crab grass. We haven't had any of these in our lawn in a few years (presumably due to the pre-emergent) but it's my "go-to" for crab grass in my neighbor's lawns (at their request) or my son's lawn.
I use mesotrione (Tenacity) primarily to "light up" poa annua and or poa trivialis in the lawn. The poa annua then gets hand-pulled. The poa trivialis gets killed with glyphosate (two applications about 7 days apart), including an extra 6-8 inches of what appears to be only "good grass" all the way around the triv-infected area.