Not sure where you are located, but it is best to buy pasture seed locally. Seed that works in my location, may be a bad choice for you, doesn't tolerate the heat or cold at your place. Asking your Extension Service for information could help you learn more about good local seeding for success methods. Read about how wonderful Buffalo Grass is!! Bought some, never even got it to sprout! Turns out that our State never gets BRIGHT enough sunshine to even get the seed warmed up for sprouting. Buffalo Grass won't grow here, so money was wasted from not knowing about the light problem. Totally an "out west" grass seed, because their sunshine is so much brighter than ours.
I went to my local grain elevator and asked them about grass seed. Told them I didn't want any of the fescue because of having foals in the future. We read ALL the labels of the available seed, to find a specific Mare and Foal mix with no fescues. It contained a variety of seeds, grasses, white clover, with little ANNUAL seeds. So this stuff, well planted, would establish a long term pasture of year-around grazing. Some plants do well in cold seasons, while others do well in summer heat. I want grazing year around, so I don't have to buy hay to feed during warm months.
I have found that regular mowing of pasture makes the grazing plants grow MUCH BETTER! You don't mow short, and you may need to mow often to keep stuff growing and being a good feed for the animals.
Prepare the ground so seed has a good place to sprout, succeed in growing. I drag and cut the ground with my disc, then walk with a drop seeder to put the seed exactly where I want it. Then dragging over the seeds for a bit of cover. I have not found just broadcast seeding, especially if overseeding an established pasture, to be much use. Broadcasting just doesn't seem to get seeds well started, so I consider it pretty wasted money.
Price of GOOD seed is VERY expensive. However if you plant it well, keep it mowed to encourage good root development, it is worth the price. My pastures now are very good, deep roots, excellent grazing for my horses. It still is producing even in the heat and no rain we often get at late summer.
Do get a soil test on your ground, then tell the fertilizer place your crop is GRASS for grazing. With the soil test they will know exactly what and how much of each item is needed to improve your field. This means your fertilizer money is well spent, not wasted on products that were not useful to your land.
I would think fertilizing and planting as soon as you can work the ground, would get your pasture going for this summer. Our dirt is very wet, so at least a couple weeks before we can do anything here in the North.