You're doing a good thing by warming up first with a walk or slower paced run. Nothing wrong with that.
The best way to improve your running, simple as it sounds, is to run more.
Consistency is your best friend. The way to be consistent is to slowly increase your training load. If you go out and run hard one day, then require several days of rest, you probably need to back off your pace or distance a bit.
Concentrate on getting into a routine. I'd suggest around 3 days per week if you're just starting out. More experienced runners can bump that up. Work on some short or medium distance runs during the weekdays and a longer run on the weekend. You're building up to the 10km, not running the whole thing right away.
Slowly bump up the distance of each of those, maybe +1 mile every week or two. The idea is to build a progressive training load that your body will adapt to. Walking part of the time is fine if you need to. You should be able to carry on a conversation. Breathing too hard to talk = running too hard. The reason is that you're working on your aerobic engine. Harder efforts get more into your anaerobic capacity which is not what you need for 10km. Hard efforts also increase the risk of injury.
Hope that helps. [source = me, 6x Ironman finisher]