New to me lawn, Let's make it better! I need your advice!

Hey guys. So I recently bought this home. It was built in early 90's and I know the prev. owner. He didn't really do anything except cut it for the past 20yrs.

So, I'd like to of course make it nicer with a budget minded approach.

Here's the current state of the yard. It's almost an acre.


As you can tell, the front of the yard near the street is at the highest elevation.



Here's the area's and their visual soil samples




Grass in area 1 is by far the best, but still has weeds.





Area 2 is weeds and just dirt....





Area 3 is a mix.





Back yard is random..





Some better photos of the front that kind of show elevation etc...






So, I know I need to Aerate and over seed for sure.
I didn't want to 2D4 because I wouldn't have much yard with grass. lol
Figured I would try and overseed and feed as much as possible this year and spray 2d4 etc... next year.
I was going to also lightly dethatch (a pull behind drag, not a powered)

I have tons of moles so I need to put something down in general to kill everything including the grubs.
Home depot has this for pretty cheap, on the right. It's $6 a bag. Only need a few bags.
Is the one of the left better for twice as much>



Im cutting with a Scag 3 wheel at around 4", also what the prev. owner did.

New to me lawn, Let's make it better! I need your advice!