I always use the old sod to make new plant beds. Make part of an old lawn into a raised or elevated plant bed by stacking the old sod upside down and at least a foot or two high in the middle and then stacking fewer to slope to nothing by the time you get to the edges.
You will be dumping a good 4 to 6 inches of top soil over the upside down sod and making a smooth bed that will be settling as the old grass and roots decompose. I'd wait for a month or so before planting shrubs, trees, perennials. Then add the finest ground mulch you can find. 2" For all your beds taking care not to cover woody stems and trunks. On shallow rooted plants such as Daphne, Rhododendrons, Azaleas only mulch 1 inch or even less.
Or if you don't know where you want a new plant bed just make a pile somewhere, turn upside down, dump left over soil on top and let it decompose into soil with great organic matter all ready to feed the soil organisms. In no way does this mean that soil has the necessary chemicals to perform photosynthesis. Always need a bit of balanced fertilizer added, I usually use half the recommended amount and applications.
You asked what to do after you remove your sod. Nothing! Yay! It is ready to install new sod, have a company come out to spray seed and mulch or do your own seeding using a seed spreader do not throw by hand. Fertilize after the first mow in 2 weeks.
Use a rotary fertilizer spreader as well. I love the Scotts hand held rotary spreader. Cheap and works great.
As for your dream of letting nature do the watering, throw that idea out. Nature is smart enough to not deal with mono crops, crops of anything. A lawn needs basic maintenance without which one should consider gravel.
Water everyday, possibly 3 and 4 and 5 times per day. Shallow watering just enough to keep the seed bed moist, never allow that seed bed to dry out until your first mow. Then you start allowing the soil to dry before watering again. This is how you train your grass roots to grow deep and when others are allowing their grass to go dormant, yours will be lush and cool and dark green.
This has worked well for me and those I taught to grow a lawn; when you step on your grass and it springs back, it is not time to water. When you step on your grass and the blades stay down so you can see your foot print stay down, it is time to water.
Water very very deeply. Soak it so the soil is wet 6 inches down. As the soil dries it dries from the top. The roots are going to grow to get to the moisture at the bottom. As your grass gets trained you will find that you will only need to water once per week. One inch of water applied once per week will save so very much on your water bill.
Never mow shorter than 3 inches. 3 1/2 inches is best. Once per week. Sharp blades. Aerate by removing plugs of soil to disintegrate on the lawn once per year. If you rake up that soil or disturb the seed bed after removing the sod you have to make sure you use a roller, a water filled roller to compact the soil to ensure no dips and hills and lakes in your lawn.
If you decide to use sod, use the roller after you've installed the sod. Look into Dr. Earth's Lawn Fertilizer, fewer applications, lasts longer, slower at first to 'green' and sorry, wish I owned stock in just that formulation, grins. I am amazed at the difference in health of my grass crops...worth the extra money)! using your Scott's hand rotary spreader. Healthy, vigorous grass can and will out compete any weed. Keep the height of that grass no lower than 3", not kidding.
MiniMe I am assuming you have cool season grasses. I sure hope so because they make a lot more sense than do the warm season grasses in terms of maintenance. Maintenance done correctly means a healthy crop that can fight disease, out compete weeds. Check the pH of your soil in your lawn bed. It needs to be at the lowest, 6.5, better at 7.0. This is done by adding lime very judiciously only when you've done a legitimate pH test or two. Your plant beds might be more acidic.
Gorgeous edges make a lawn. As long as the grass is a uniform color and texture, it will be the edges you see. If those edges are consistent radius before changing the direction and radius point, that lawn will be stunning. Make a little trench 6X6 inches or 8X6 inches...use a small flat shovel to define the edge of the lawn. Throw extra soil on the beds. Do not worry about moles and hills. Those little animals are doing you a huge favor. Just knock the hills down, sweeping the soil thinly into the grasses. They aerate and eat grubs, larvae that are very damaging to a lawn crop. I charged people money to do the same work; called top dressing, and aeration of the seed bed.
I have been assuming much. Please add more info and pictures so we are better able to answer your needs.