Gardening Novice looking for some Advice

Hi guys, new to the forum and completely new to gardening so a bit clueless despite hours of searching online.

I have recently moved house and have a fair sized garden (total area 170m²). This comprises of mostly grass apart from one square section (approx 12m² that had been dug up and replaced with bark as a kids play section).

Not after anything fancy here, just a good quality lawn with no weeds! It was absolutely infested with all sorts of weeds but mostly dandelions - I dug out several hundred dandelions a fortnight agowhich was made easy with my new weed remover but it has inevitably left unsightly pot holes everywhere. I then sprayed the whole area with selective weedkiller in a pressure sprayer.

My plan is as follows:
-One week from now now the lawn really short
-aerate + scarify
- fill the small holes with topsoil and also place a thicker layer of top soil in the barren 12m² section
-spread grass seed with spreader
-water regularly (have purchased a garden sprayer attachment for hose) and cut grass to correct length once its growing

Main concerns and areas I am unsure about are:

- type and quantity of top soil I require - seems this can get very expensive

- what to do with the soil cores after aerating and can I spread the grass seed straight after this

-how best to prepare the soil in the bare area prior to seeding

Any help would be hugely appreciated

Can post pics if that helps