Your local university extension office, USDA Soil Conservation office, or farm fertilizer company can give you a worksheet on how to collect and submit a soil sample. You usually write on the form what crop(s) you want to grow and the test results will come back with recommended application rates. If you have a USDA office it would be most helpful to you to stop in and ask them exactly what you've asked here, since they'd be most familiar with your local soil and weeds, and they might even come out and take a look at your field if you ask for advice.

Lime can be applied any time it's dry enough for the delivery truck not to damage the soil by compaction, but they're busiest in the spring and aren't always available to deliver at your convenience. It takes months for the lime to have any effect on soil pH in the root zone, so you apply it as much in advance of planting as possible. The truck has a broadcast spreader on it and they bring however many tons you order, you point out the area where you want it. Some companies also have smaller broadcast hoppers you borrow to pull behind a pickup and spread it yourself.

You already said you were going to start with a soil test. That should help tell if you REALLY need the lime. If yes, they will mix lime in the fertilizer package you order. Don't buy more stuff than you need to, or minerals you DON'T need. That is why you test soil, don't guess what is needed for your crop, pasture grasses. The test results allow fertilizer folks to assemble the correct mineral proportions and quantity that you need for this dirt.

I am at the point now where I need to just add lime to my fields yearly, have fertilized according to soil tests for 3 years. We really put a lot of nitrogen in the first couple years, spring and fall. I have tested each year so know the soil is changing. I will keep soil testing, probably every 2nd year now. I do soil tests on each field, they are not all the same. Other folks don't test as often, but I have hard use on my small acreage with the horses, so I want to keep my good pasture in high production. Soil is now in good heart, lime needed is the aid/helper link, to allow plants to use nutrients they already have in the dirt.

I would start with the bush hogging, shorten the weeds way down so you can see what you have. Probably then I would just go on with a complete herbicide overspray to kill everything, then drill in the grass seed and fertilize.
We have a local county No-Till grass drill available for rent from the Extension office. You might ask your local office about machines to rent when you give them a call as advised above. Does take a fair sized tractor with hydralics to pull our drill. No-Till saves the dirt from being uncovered, exposing all the weed seeds to germinate, erosion. Working up the soil, then drilling in seed, is second choice method. Drilling in makes for a better crop than third choice, work up ground, broadcast seed with fertilizer, and then lightly dragging over it.
Watching the weather closely, I have everything in field ready for when rain is predicted a day or so later. Race like crazy to get it done so rain can then get the fertilizer and seeds wet, working quickly in the ground.
I rent the small wagon/cart/hopper from the fertilizer plant, very cheap, holds a lot. Runs off the tractor PTO and spreads fertilizer very well. You could mix in the grass seed with fertilizer if you have worked up the ground, spread everything on one pass. I have a small chain harrow to drag with, cover the seed. Just broadcast seeding, without good prepartion or as an overseeding, doesn't seem to work well for me. Waste of seed and time.

Buy good seed, check the contents to know what you are planting. Avoid Fescue grass for horses, maybe also for cattle. Affects bred mares deliveries of foal.

You have until about mid-Sept to get everything done, make use of the early season fall rains to get seed going without drowning it. You should have a fair crop of new grass, good roots by the end of Oct to survive winter. Don't plan to use for animals until next summer. Needs to get well established before they start tearing it up.

If you aren't in a big hurry, you might want to use a few goats to upgrade the land. Tasty weeds in one end and top notch organic fertilizer out the other.

Lynda