A 10k run is a super distance to run - much more reasonable than 42!
You'll still have to train for it though. I don't know where you're starting from fitness wise but I think you'd want to start out running 3,4 or 5 k. You want to do it 3 or 4 times a week and slowly build up your mileage. I think you're going to want to use one day a week to lengthen your runs - usually Saturday or Sunday. You can still do your 4k runs in the week but then on the weekend you want to try and stretch yourself out to 10k eventually. Take it slow...our bodies simply aren't used to running and in the beginning you'll have various bits of pain in your hips, knees, shins, calfs etc. It can take a bit of time for your body to adapt. Invest in a good pair of running shoes(trainers as you say over there!).
You might want to start tracking your pace and give yourself a goal based on that. I think for a lot of beginners a good goal is to run that 10k race in less than 1 hour. You may be faster or you may be slower and it really doesn't make a lot of difference. The main thing is to get out there and to push yourself a bit. I did a 10k run in 44 mins last week and that felt like a great accomplishment for me...first time I went under 45 mins.
It is a terrific fat burner. I'm running so much these days that I can literally eat whatever I want and not gain weight...burgers, fries, poutine, pizza, coke, sugar in my coffee, ice cream.
Do it!!!