I've had years of experience with 'lawns'...the grasses I am familiar with are lawn grasses that do well in the northwest. From your picture, which looks oh-so-familiar, I would do one of two things: #1. Aerate, pulling plugs of soil out of your lawn and leave the plugs on the lawn; edge your lawn using a weedwacker or metal-blade edger between concrete and lawn; fertilize with 'Dr. Earth's Lawn Fertilizer'...it is 'organic' and thus extended release...follow directions, it takes more but lasts a lot longer than 'Scott's' or other inorganic fertilizers (I was amazed at the results with Dr. Earth). Be patient, it takes a week or two to begin to see results. Mow so that the grass is NO SHORTER THAN 3". The NW grasses have huge root systems that NEED the 3" of top growth to support properly and become vigorous to compete with your weeds so I am assuming that your grasses are similar to the NW's. Water deeply and do not water again until dry...you should be able to see your footprint, the grass blades stay down when you step on them when the grasses need a deep watering again. The water should reach at least 6" deep into your soil. Your goal is to create a drought-tolerant grass as the roots are forced to grow deep to get at the water. Maintenance will be 1" of water per week (place cat food cans or other straight sided containers around your lawn while you irrigate, use a shovel to see the depth of water into the soil. Never water everyday, never by hand). If you have automatic irrigation, hopefully your lawn is irrigated separately from your planting beds, check each zone with your cans...15 minutes should yield 1/4". If your soil is wet to 6" 15 minutes per zone will do. Don't water until the soil is dry and your grass provides footprints. Your goal is 1" of water once per week. You can see that you will have to be vigilant. You want your grass to get deep roots but not be water-stressed.
Check the pH of your soil. Use your cooperative extension service for an accurate reading. If below 6.5 or whatever your service recommends, use lime to raise the pH to 6.5 - 7.0. This takes time to happen. Test again. Never add lime without KNOWING your soil pH.
Your grasses should be thriving (as well as some of the weeds...shallow rooted weeds will be stressed by the deep watering/drying out and the grass is strong enough to compete with the rest), here I would use broad-leaf herbicide. You need to dampen the grass and weeds so that the herbicide can cling to the weed-leaves. Don't fertilize with herbicide. You shouldn't need another application for months...I think you will want to stay with extended release, organic fertilizer anyhoo.
For crabgrass, in a small lawn, I would dig the stuff out and/or use glyphosate. Use rubber gloves and hold a small amount of glyphosate (Roundup) in a dixie cup. Wet your thumb and a finger in the glyphosate, don't DRIP, and wet a few leaves of crabgrass by pulling them through your glyphosate-dampened fingers. Be patient. It takes 2-3 weeks for the product to kill the roots. You WANT the crabgrass to be growing in order for the glyphosate to be able to kill the plant.
Meanwhile, our eyes SEE the edges of a lawn, not the stuff that makes it green. Make sure your edges stay trimmed. Don't mow or weedwack the grass along your edges shorter than the 3". Other than the edges by your sidewalks/driveway, your edges for your lawn should have a set radius that only changes when the edge goes the opposite direction. Like a highway...no changing the radius until forced to go another direction. Use a smallish, flat bladed shovel to clean up your edging, make a trench to collect water and throw the soil back onto your plant beds (hopefully they are 6" to 1' above your lawn). If you've got bark on your beds (please, I hope there's no 'landscape fabric' beneath), think about raking it up a get rid of it in your greenbelt or an older, established part of your garden. If you've used plastic, GET RID OF IT. Find a place that makes DECOMPOSED organic mulch from human poo. It is beautiful, dark taupe, fine-grained, no rocks, sticks, no weed-seeds, no pesticide residue...and smells wonderful, HONEST. It is also, the only organic, decomposed compost that is TESTED. Not for your vegey garden because of heavy metals, this recycled human poo and sawdust is thoroughly decomposed, tested at least 5 times and no longer resembles in anyway the stuff it started out as. This stuff feeds your soil, encouraging micro and macro-organisms that your ornamental plants need for uptake of nutrients, drainage, pH stabilization...healthy, live soil is essential for plants. Place 2" deep, keep away from bases of trees and shrubs and keep to a depth of only 1/2" on top of shallow-rooted shrubs (examples: azaleas, rhododendrons, daphnes). In my career, I refuse to use non-decomposed bark. It is creating deserts in our urban landscapes.
The other thing (#2 solution) I would do would be to rent a sod-cutter and get rid of your front lawn completely. (Now is a great time to make a new plant bed...use old sod, turn upside down, top with good topsoil, rake smooth, plant trees and shrubs or let decompose, settle and plant the next year...use human poo mulch to prevent weed growth). Get decent top soil delivered, spread it (1-2") on your lawn area, roll it with a water drum (crucial), lay new sod (keep moist until rooted about 2 weeks), mow on high (3"), fertilize with Dr. Earth or other organic lawn fertilizer with michorrhyza and bacterias, water deeply and allowing to dry out training your new grass to develop deep roots. Your grass will be able to out-compete any weed and all you will have to do other than water deeply, mow high, fertilize with organic, slow-release fertilizer, edge with line trimmer (weed-wacker) is aerate once per year. And be humble as the compliments come in...grin.
I hope this helps...if you have a bad thatch problem (+1" of non-decomposing grass matter between blades of grass and soil), the #2 solution would make total sense. Cut out your sod as deep as you are able.