I love your question Joe. Most people who are able to afford to will OVER DO everything. Plants and soil are nothing like humans/animals. Don't fertilize until after the first mow. Did you take the sod off with a sod cutter or are the roots still there? You need to rake, grade, ROLL with a water filled roller, more grading filling holes and roll again. Use a rotary spreader for the seed. Rake with the back of your rake. Roll again. Water 3 or 4 times a day just long enough to keep the top 1/2 inch of soil moist. If your grass hasn't grown enough to be mowed within 14 days, relax and do it again in the spring. Better than seed is sod. Check the prices...you might decide installing sod is worth considering. Fertilize after the first mow...but if it is fall, keep the nitrogen low. Find a product with Nitrogen as the lowest of the three numbers and better yet, find an extended release organic formula made for fall...check Dr. Earth's Lawn Fertilizer. Worth every penny...
Gotta tell you, when we had extra sod we would lay it out over the gravel in the yard and it grew just fine! Do not worry about amending your soil. Get a soil test through a cooperative extension service. At least test the pH. Getting that lawn bed graded and 'compacted' is critical...and grass loves a bit more alkaline of soil. Preparing the lawn bed, creating the edges (6inch by 6inch trench) using consistent radius, seeding over raking, rolling!! Using the best seed you can find with zero weed seed, no peat moss...hey check out grass spray companies! Perfect. They come out and spray great seed with a bit of fertilizer and a weird mulch to keep the seed moist...they do a great job. They will tell you what to do and when, they have to make sure their product sells itself.
For maintenance, you have to read some of our answers on lawns. How to train them to be drought tolerant and save you lots of money on water, watering deeply not watering until you see footprints on your grass, how to keep your lawn weed free by mowing your lawn no shorter than 3 inches. And lots of other basics anyone who wants to grow a lawn should know. Without spending lots of money!
Notice in the two yards that one is unable to see the entire yard from the back door. There is peninsula bed that breaks up the view. Now for kids, whilst they are younguns, the plant material will be too low and new to obscure your view of your kids. Just something to think about as your kids grow up fast! To see your entire back yard from the back door is boring. Something to work on as they grow. They would LOVE the gravel...a few boulders. Heaven for kids. And I have to warn you to never put a play gym out near the green belt, if you have one. Keep your play gym close to the back door. There are cougars, bobcat and bear that come into town. The cats will stalk you and figure out your schedule. Always be aware.