First-weed problems

Hello all - medium-term lurker, first time poster here - hoping to tap into the fountain of knowledge this excellent forum provides!

We're FTBuyers of a lovely house with an equally lovely south-facing back garden, but my green finger experience is nil - however I'm now fully invested in learning the ropes to cultivate a beautiful space.

The grass seemed a good place to start, which I'd say was about 70% weeds - so purchased Westland All-in-One Aftercut and set about spreading over the offending areas. As you can see, I believe it's doing its job with the copious black areas....but now I'm not sure of the next step.....

I've trawled online but cannot find a definitive answer - do I leave these unattended to allow, in theory, the grass to grow where the dead weeds were? Or I do I need to physically remove them myself? I'm also unsure of watering and mowing protocol at this point.....

I'd also like to sort the two bare patches just about visible (top-right and middle-left) left by previous owners' heavy items - is it worth just seeding those areas, or the whole lot given the state of the grass across the board?

These are probably all very basic questions - but any advice much appreciated, many thanks!