Hello, johnmang99.
It sounds like your yard needs some work and you have some good ideas. What you do to help your soil and grow good grass depends on what kind of soil you have and the condition it is in.
You mention several actions that normally would be used for specific purposes and/or to correct specific problems.
Core aeration is normally done in the Fall and sometimes in the Spring. It has two benefits. First it helps to loosen the soil. Second it brings microbes up from below the surface of the lawn... the microbes help decompose any thatch you may have on the lawn. From your description, thatch is not a propblem for you. If your soil is not compacted, you could wait to aerate in the Fall and spend your money and time better in other ways to get your lawn going.
The only reason you may need to apply lime is if your soil pH is below about 6.0. Otherwise lime addition will have a negative effect on your efforts to grow grass. In this regard, soil testing is very beneficial. Check with your land grant college to see if they offer the service and how to prepare the sample... or check out one of the sites below for a simple soil test kit. If you don't want to do this, I would hold off applying lime.
Outsidepride.com
Soilperfect.com
I would recommend renting a seeder which slits the soil and plants seed in the slits. This would be very effective in getting good grass to start growing throughout your lawn, in the bare spots and in the areas where weeds and crabgrass are growing. (Don't try to get rid of them yet.) The earlier in the Spring that you plant the grass seed, the better chance you have of it doing well.
After seeding, apply a Starter fertilizer. I generally recommend
Scotts because of the consistency of their product throughout each bag. Apply at the recommended rates using spreader.
Grass seed needs to be kept moist and the seeder doesn't plant it too deep, so initially water twice a day (unless it rains) but only apply enough water to dampen the top 1/2 inch of the soil.
The seed should start sprouting in about 1-2 weeks. After about 5 weeks it may be ready for its first mowing. After mowing the new lawn 3 times it will be well enough established for you to attack the weeds and crabgrass.
For this I would recommend that you purchase a lawn treatment applicator which will attach to your garden hose.
Ortho Dial-n-Spray is a good one that is easy to use. For the weeds I use Ortho's Weed-B-Gone Max for weeds. For crabgrass I use Ortho Weed-B-Gone for Crabgrass.
My personal preference is to purchase concentrates and dilute them myself... much less expensive. The Dial-N-Spray dilutes the concentrates for you. To hand spray weeds and crabgrass (you'll get to a point this is all you need to do) dilute the concentrates in a one-gallon milk jug according to directions and then pour what you need from a milk jug into a hand-spray bottle. (Buy a high quality hand spray bottle in the cleaning section of your local hardware store. Don't try to use an empty Windex or Fantastik bottle - the sprayers won't hold up.)
Another excellent product that attacks weeds and crabgrass is the
Bayer Advanced All-In-One Weed Killer for Lawns which kills both weeds and crabgrass. I have used this product and still would if I could find it in local stores. I think it does a better job against the crabgrass and against the other weedy grasses as compared to the Scotts product.
A note about weed killers: since I don't know where you live (not in your profile yet) be aware that northern and southern climates require different timings, different types of seed and different weed killer treatments. Read directions of all products and talk to your retail specialist in the store for advice.
The first application of weed killer may not kill all of them. Some may be just germinating when you make your first application. Don't get discouraged. This is multi-year process.
About the third week in June make your second fertilizer application. If your lawn or your neighborhood has seen grub problems in the past, then use a fertilizer with a grub and insect killer. Otherwise, just use a plain fertilizer such as Scotts Turf Builder.
Around Labor Day, apply fertilizer again, followed by your second weed and crabgrass treatment.
In Ohio, where I live, September is a good month to aerate and to overseed our lawns. You may want to plan to do those two things about that time, too. Just be sure (read directions) that you separate your weed killer application from the overseeding with enough time.
Finally, in late October make a final application of fertilizer... use regular Lawn Fertilizer or, better yet, one formulated for this late application (e.g. Scotts Turf Builder Winterguard). This application is the one that will give you a thick lawn by promoting root growth. A dense grass root system will help reduce future weed seed germination and growth.
By following this plan you should see your weeds and crabgrass significantly diminish even this year. Next year your yard will look much better, and you can start in March or early April with the
fertilizer plus crabgrass preventer application.
Good luck. Feel free to contact me if you have questions.