If you're arriving at King's Cross, you can leave your luggage there - there's an Excess Baggage Company branch in the main concourse, underneath the Parcel Yard pub (to the right of platform 9 - walk past the platform 9 3/4 thing and you'll find it!)

Then you can quite easily head to Borough Market via the Northern Line to London Bridge (or the number 17 bus from the Caley Road) or Covent Garden by the Piccadilly Line, or just on foot.

Don't take the tube map as being an indicator of how far things are from each other - it's a diagram and no more!

From King's Cross, you can either head direct to Heathrow via the Piccadilly Line (takes max 1h, not the most comfortable, but dead easy) or via Paddington (take a Hammersmith train to Paddington - there are lifts and escalators on an indirect route at Paddington, although I usually use the steps).