Admiral,
I do a lot of indoor karting (hire karts) and unfortunately there's no single correct way that works with all karts at all tracks.
Some karts have drive to only one rear wheel, some to both. Sometimes the brakes are good, sometimes non-existent. Some karts require left foot braking, some right.
Indoor tracks vary considerably in the amount of grip they provide in various corners.
Example: The last time I raced the track had a right hand hairpin after the main straight which was slippery as hell right up to the apex. After the apex there was an enormous amount of grip amount right upto the following corner. It was so slippery at the end of the straight that just touching the brakes with the wheels pointing into the turn put the kart sideways. Approaching wide I tried to brake really late and hard just before the corner (and keeping the throttle half open) sending the kart into a sideways slide towards the apex at a totally suicidal velocity threatening to send me straight into tyres. Fortunately the kart bit in

and I could floor it in a smooth line out of the corner.
This method doesn't always work - actually hardly ever - but it gave me over a second on the next place man and 0.5 slower than the lap record, not bad for a hefty 93kg guy

. I also ended up with major bruising on my left side
It helps to know where the grip is greatest and that may or may not be the normal racing line. Generally the best technique is to drive smoothly turning in a bit later rather than earlier and moderating the throttle i.e. not taking your foot all the way off throughout the corner and braking lightly with your left foot all the way to the apex.
I also use my weight (considrable!!!) - leaning forwards on the brakes, leaning to the side around the corner and bending backwards to help traction out of the slower corners. (Read karting explosion thread)
Remember: you will never be quick if you are slower out of a corner than you were into it.
Happy karting...