I don't think you need to go out straight away and get specialist running shoes for £70+, so if you don't have some comfy trainers already I would start in Sports Direct or JD. For running on concrete paths you need something with good cushioning to absorb the impact, so comfy and squishy is what you are looking for. If you find you enjoy it and want to stick at it, then yes it is a good idea to get some shoes fitted and spend a bit more to protect your feet and suit your running style. Likewise if you decide you want to do trail or fell running later on there are shoes specifically for that that have grip, a lower profile and less cushioning.
Parkrun is free to attend every Saturday and they record your time if you print out your barcode, so you can see how you improve. You can also walk/jog/run at your own pace and no-one minds. It's very well organised and they have many locations up and down the country.
If you are going to run in the evenings I would recommend you get a cheap pair of running tights, light weight rain proof jacket and some sort of reflective band or vest so that oncoming traffic will notice you. A sports bra is a must too. In this weather I just wear a running vest or short sleeved tshirt under a running jacket and I am warm enough (I get VERY hot when I run). When it is bitterly cold you might want to take gloves and a hat too.
To plan a route local to your house you can use a tool like geodistance.com who let you plot out a route on Google maps and measure it in miles or KM. Don't worry if you get out of breath quickly, just have a little breather and walk when you need it. Eventually you will be able to run/jog further before you need to walk, and after a while you will find a comfortable pace and not need to stop much or at all. If you prefer a rigid structure to stick to then couch to 5k will probably suit you too.
There are a few apps you can use on your phone to track your runs like Endomondo or Mapmyrun, these are nice to record your routes and compare how you are improving from one week to the next.
Anyway, hope you enjoy your first run!