What an exciting project! I got totally hooked gardening in my first house, and now it’s my favourite pastime. This despite it being a new build, with a back garden that was full of builders rubble and very hard to work with.
Personally, I’d get the concrete up then assess what is underneath. When you break up the concrete, don’t be tempted to dig the chippings into the earth and definitely do NOT add more. Often, rubble in the soil aggregates over time and causes a virtually impenetrable layer, causing significant drainage problems. Grass likes to have its roots in soil, and ideally you want a good depth of good quality topsoil without stones - at least 4 inches - and ideally beneath the topsoil you won’t just have a layer of hardcore, you’ll have earthy subsoil. If you do have builders rubble or hardcore under the existing topsoil (once you have revealed it beneath the concrete) you can still try laying grass, but you will find it is harder to establish due to the drainage issues.
We took up a vast layer of ornamental stones in my current home (previous owners were old people with dogs owned the house, they just tipped stones everywhere!) and we spent a lot of weekends manually sifting out the small ones. Then we left the soil for a few months before laying turf and I’m glad we did, as it was full of brambles which emerged and we were able to fully dig out these and the major weeds before we put the new turf down. Also you don’t want to be laying turf in August (research good times of year to lay turf).
Look at your immediate neighbours’ gardens. Do they have successful lawns? If your house faces north, you may find it is fully shaded by your house in winter so grass will become damp and mossy, but in summer that shade will help a little to protect it from the drought and searing heat until the hottest part of the year when the sun is overhead. If your garden is south facing, then you will find the lawn needs help earlier and longer in the year to protect it from drought (letting it grow long is the best thing).
Regarding the borders, are the current borders thriving? You can dig down to sample what’s underneath, it will give you a guide what you’ll find under your concrete - if the existing borders sit over the top of a layer of rubble, it’s a guide that the concrete likely does too (although gardens can be rubble in some areas, earthy in others, depending whether the builders had to level off the land to build). I have grown very successful borders over rubble, but it was so free-draining it severely limited plant choice. Hopefully it’s a problem you won’t have.
And again look at your neighbours all around the area to see what is growing well. If you see a pretty garden, compliment the owner and ask about key plants if you aren’t sure what they are.
im such a fan of ripping up concrete and letting the earth breathe again through its green lungs. Hope you have lots of fun and success.