Early fall is the time to seed cool season grasses. Sounds like you will have moved by then. Just keep the grass/weeds cut in the meantime.

I would not recommend compost and seeding. If you're moving it seems a waste of money and your time. I agree with Becky, just mow and trim.

I should have said that my main goal is to make the yard look nicer so the house will sell more quickly, and possibly increase in value. If I can't enjoy the new grass for long, that's okay with me :)

If I planted some fescue now, would it germinate this spring? Would it survive the summer?

New grass will germinate in the spring, but so will the summer weed seeds like crabgrass. And no, spring planted fescue likely will not survive the summer. By July it will be all crabgrass.

What I would do is to apply organic fertilizer now at 10 pounds per 1,000 square feet. In April spot spray the weeds in the front with Weed-b-Gon or Weed-b-Gone Chickweed, Clover, and Oxalis Killer or equivalent. If that leaves bare spots in the front, then buy a few pieces of sod to fill the gaps and holes. Then in May apply organic fertilizer again at 15 pounds per 1,000 square feet. You can use alfalfa pellets, soybean meal, corn meal, or corn gluten meal as your organic fertilizer. These materials will work much better than compost for a small fraction of the cost. Here's a picture showing the effectiveness of organic fertilizer.

Thanks to mrmumbles for posting this picture several years ago. He dropped a handful of alfalfa pellets on that spot in mid May and took the picture in mid June. You can see the improved color, density, and growth. It does take 3 full weeks to see the improvement with organic fertilizer. It is slow to get started, but very effective.

Also mow at your mower's highest setting and start watering deeply and infrequently according to the following guide. Proper watering the most important thing you can do.

Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.

Thanks, everyone!