Hi Jo, Welcome to the forum.
Take a look at the plans on here or alternatively google: couch to 5k. Any of these are good for starters and will get you building up your mileage gradually.
I'd recommend getting a good sports bra (but you probably have one of those already for kickboxing) and just take it easy! Don't try and overdo - just stick to the plan and it'll see you through.
Regarding the other sport - i'd keep doing it. On rest days - you can go kick boxing, cycling - whatever you want. I normally use my rest days from running to cross train (cycling, swimming etc). Just carry on as normal with kick boxing and build up your running nice and slowly.
How quickly to see results? How long is a piece of string?
It really depends on so many differing factors that it's not possible to say exactly. Get to the end of the program and see how you feel. I've heard between 1-4 months but that's a complete average. Your body is unique - so it could be quicker or slower.