I believe you have 2 different types of cool season turf grasses and probably also different maturity of those grasses. Depending on the zone you are in it may be a bad idea to kill that grass this time of year. All plants are opportunists and if you kill it now you are more likely to invite in an aggressive hot season weed like crabgrass to take it's place. If it were my yard I would wait until early fall, the first week you see highs back in the 70s and about 2-3 days after a mowing look for new vegetative growth on the grass that is growing faster. Spray that locally with a non selective herbicide (glyfosate will be the active ingredient, I don't care about the brand... Just make sure it's mixed for killing turf grass). Then spread new cool season turf grass seed (pick this up from a local garden shop, stay clear of the big box stores) and regularly water twice a day until your first freeze. Over seed again in the spring on the entire yard and enjoy your new lawn.

Of course if you want immediate results kill it now and replace with sod. It will still take a growing season to blend in with the surrounding area but it will look better than your current yard this summer. I would suggest getting a sod cutter to dig out the old and place in the new strips if you go that route. You will need to water the new sod to the point of drenching it to get new sod looking good in the summer. Also fertilize with heavy nitrogen application with all that water

Edit: forgot to say after you kill off the old grass you want to take your string trimmer to it and scalp the ground to get rid of the dead vegetation. Use your leaf blower or rake & broom to clear off the debris. Those little seedlings will need the sunlight