Ok, I will summarize what I found and am going to do and you let me know if that sounds OK. BTW, I used Home Depot's "Lawns 1-2-3" book and seems like I know too much about lawns now
1. I did identify soil that I have. Seem like it's clay. Fertile in general but problems with compaction.
2. I did identify that I have cool season grass right now. Didn't really go into what kind, was too much

3. I did identify weed on a picture. It's an annual bluegrass. Need to apply preemergence herbicide NOW.
So, right now I already took samples, taking them to my local university extension. But judging by what I found - most likely I will need to apply a lot of lime..
My plan for this fall in steps:
1. Cut grass really short, like 1-1.5 inch tall.
2. Do core aeration
3. Do top dressing with a good organic matter. Compost, etc. Around 1/4-1/2 inch layer. Best if spread and fills plug holes.
4. At the same time I will do overseading.
5. At the same time do slow release fertilizer
6. Lime if needed
Then water every other day or so.. Apply preemergence herbicide to control weeds.
Any suggestions on my plan? #2 and #3 sounds like a hard labor.