Answer these:
Where do you live
Do you have a sprinkler system
Is your lawn mostly shaded or in the sun/ how much
What kind of grass do you have and what kind do you want
How many hours a week are you going to do lawn work

And who owns that slope, you or the neighbor?

What I see is dripline damage. Water dripping off the house is pounding that soil and compacting it. People who know me on this forum know that I would not use the C word loosely. It is almost impossible for a home owner to compact soil, but this is one way. But ignore that because it is inconsequential to your other issues. You MUST get the drainage fixed on that side of your house. There should be a low spot exactly at the bottom of the hill. That low spot should be at least 3 inches lower than the soil at the edge of your house. In other words you need to remove several yards of soil to correct that. Fixing that will correct some other issues I see.

That low spot along the bottom of the hill is called a swale (nicer term than ditch). It will allow rainwater to drain out to the curb in front. Hire a landscaper to fix that for you. He should be able to do it in a morning with a tractor. When it is right the soil will slope away from the house to the swale and the swale will slope out to the front. Then you'll be starting over with new grass seed in there. Plan this job for August or September (depending on where you live).

What is a push mower? Do you mean instead of a riding mower? Does it have a motor?

In addition to botanicalbill's questions, what is your watering regimen? How often and how long do you water?

Botanicalbill, here are the answers to your questions:

Where do you live? North Alabama (near Birmingham)

Do you have a sprinkler system? Unfortunately, no.

Is your lawn mostly shaded or in the sun? This depends. The front and much of the back is in the sun, but the side is mostly in the shade.

How much? I tend to work 10 - 12 hours a day, so I would only be guessing (perhaps six or more hours).

What kind of grass do you have? It's a sickly mix. I think Bermuda and Fescue were mentioned by the guy who sprayed the weeds.

What kind do you want? Something thick, dense, and green that even I can't kill off--low maintenance and healthy works for me!
How many hours a week are you going to do lawn work? Right now, I only have the the weekends to be able to put in any decent time. Then, I usually do about four or five hours of work each day.

dchall_san_antonio here are the answers to your questions:

And who owns that slope, you or the neighbor? Unfortunately, I own both slopes.

When you were talking about dripline damage and needing a swale, I actually have a big drainage ditch (that ain't no swale!) that runs along the front of the yard. You can't see it in the picture, but it is right before you get to the mailbox and runs horizontal along the front of the yard. Would you still recommend a swale?

What is a push mower? It's a walking lawn mower that is not self-propelled.

What is your watering regimen? How often and how long do you water? I didn't know I needed to. The other month it rained for almost two weeks straight. Do I still need to water with that much rain?

The swale will run from the back yard, along the bottom of the hill, all the way out to the front ditch. The job of the swale is to carry water away from your house. Right now the lowest spot on the side of your house is that dripline damaged area. The low spot needs to be as far away as possible from the house. That would be the base of the hill.

Supplemental water is needed to carry the grass from rainstorm to rainstorm. But if the grass is not looking stressed, then you don't need to water. In your case, the grass is already gone from not watering so there is not much left to be stressed. Generally with temperatures in the 90s, you should be watering once a week. The amount to water is a full inch as measured by cat food or tuna cans. Get an oscillating sprinkler (with the turbo technology as opposed to the mechanical linkage type) and a good hose (Any Sears Craftsman is good and unconditionally guaranteed for life). Put the empty cans in the yard and time how long it takes to fill them. It will likely be longer than 5 hours, so don't just hang around and watch.

The other month when it rained for 2 weeks is when your crabgrass seed germinated. Thank you Mother Nature. But you set it up by allowing the grass to thin out making way for the crabgrass.

As for what kind of grass, I believe you are in the southern side of the transition zone. That means the southern grasses should grow fairly easily and the northern grasses might be a slight challenge. Look around the neighborhood to see what looks good to you. Find out what grass it is and we can help you get it going. Don't dally on that task. It is getting late to spread seeded bermuda if you wanted that. But for any other kind of grass, that can be done any time until October.

Thanks for clearing up the confusion about the swale! I'll look into having this done. It may not be too soon, because I have almost $2,000 in tree work that needs to be taken care of next week--three trees removed, stumps ground, etc.).

After looking around the neighborhood and looking up the different types of grasses recommended for warm seasons (Bahia, Bermuda, Centipede, Fescue, St. Augustine, and Zoysia), I tend to like the look of the lawns that look like turf--both the Bahia and the Improved Bermuda. I think I'm leaning more towards the Bermuda, because I've heard that it tends to spread rather quickly.

Okay lots to say.

When they grind your stumps, have them remove all the wood chips. You can scatter them in the garden somewhere but DO NOT BURY THEM. It will be bad enough that you have wood underground. You don't have to add more wood underground. And see if they can grind it down at least 10 inches. You want as much soil over top of the unground stump as possible. If they can remove the stump entirely, you'll be a lot better off. Grass does not like to grow over old stumps.

All the grasses you mention should look like turf. Some are easier to care for, though. Of the grasses you mention, only fescue, bermuda and St Augustine are "normal" and easy to care for. Finding experts for bahia or centipede is going to be hard. There are zoysia experts around but they're few and far between. Fescue is not recommended for warm seasons although the Marathon dwarf variety is extremely common in posh Southern California locations. They have to water far too much to keep it alive. Sometimes zoysia is easy to care for, but I consider it fragile. Centipede is extremely easy to care for but most people ignore the advice. The only thing you have to do with centipede is mow. It rarely needs water and NEVER needs fertilizer. Most centipede is killed with kindness. Also it is full sun as are bermuda, bahia, and the zoysias that do well. In the full sun you could use any of those, but on the side yard you would need either fescue or St Augustine. Or you could simply mulch the side yard with several inches of shredded tree mulch from a tree trimmer.

Thank you for the advice and details dchall_san_antonio! After reading the pros and cons you posted, I think I'll go with the centipede. Hopefully, I will not kill it with kindness the way I sometimes do my plants!

One more question: What is the best way to get centipede started? Should I kill off the crab grass mix, dead spotted one that I currently have or put out seed on top of this and water?

One thing to check before you decide on centipede grass. Get a professional soil test done to determine the soil pH. You cannot get good results with a home test kit. Those things usually end up testing the pH of your tap water. Centipede is only good in acidic soils. If you have an alkaline based soil, it won't work at all.

Regardless of what you go with, you cannot apply over top of the current grass. At least you'll have to use something like a slit seeder to cut through what you have and get the seed to touch the soil. There are a couple other ways to get going, but let's see if your soil is going to be good for centipede first.

Thanks dchall_san_antonio! I'll drop by the Cooperative Extension and get the soil sample boxes. It'll probably take a few weeks to get the results back, but I will make sure to post the results.

Thanks again for all your help!

I don't know what your county agent soil test costs, but for $25 you can get an incredible soil test from Logan Labs in Ohio. You don't need all that, but if your county test costs $20, you might want to spend the extra $5 for a Logan Labs test. From what I read you get the results back from LL in about a week or less.

Thanks dchall_san_antonio, I'll keep Logan Labs in mind! The local cooperative charges $7 for each test, and they send it off to Auburn University (hence the wait time); but I'm not sure if they provide the same level of analysis as Logan Labs.