Are you mowing short and raking up the clippings?
Clover is not a problem for us as we do not mind the lawn containing anything else besides grass in it (other than a few weeds like certain thistles and poison ivy).
We use a mulching lawn mower and do not remove the clippings and also mow as long as we can (this feeds the lawn and helps suppress weeds).
Your soil there looks sandy or more mineral based which means that compaction or aireation may not be an issue. If it does not hold much water you can apply a small amount of clay, but basically I'd just leave it as it is.
From the large green spots I would say that something is making extra deposits on your grass or perhaps that is where #2's got watered in a lot? I do not know.
I never recommend fertilizing a lawn as that means more work mowing and watering. Just keep it regularly mowed at a fairly high setting on the mower and let nature do it's thing. Your regular mowing will self-select for the grasses and other plants that can survive your treatment and weather. The reason I say this is that if you have dogs as pets you will not ever have a nice and perfect lawn. Accept that it will be less than perfect and find happiness in not caring. Animal control, weed control, watering and basically not treating it like much of a usable space will make it look picture perfect, but otherwise, reality is that the lawn will eventually get weeds, get a bit beat up from animals or people playing...
We have deer making deposits on our lawn in areas and it looks much the same with spots of green extra growth. Beyond that we also have a lot of diversity of plants in our remaining bits of lawn and whatever the rabbits, deer, groundhogs, etc. eat of it does actually provide some weeding.
Specific garden beds will need specific questions. Different plants and locations may need different considerations.
At least having been there a few years you can see how the lay of the land is and how your water flows after heavy rains. Those are good observations to keep in mind when thinking of future gardens.
If you plan on growing vegetables put the vegetable beds as close to the house as possible (not wanting the light being blocked by much) and likely you will also want to plan a suitable fence to keep animals/dogs out. 6ft is minimum here to keep deer out - fine enough mesh to keep rabbits and other creatures out too. The closer to the house a vegetable garden is the more likely it will be taken care of and problems noticed and addressed.
Good luck. :)