When I start my 2002 Mini Cooper S JCW edition, right as it starts it either loads up, or hesitates for 1/4 of a second.
That 1/4 of a second has now increased to 1/2-3/4 second that it hesitates.
It also seems to want to bog down, or stumble for the slightest amount of time when it is at idle and I push the gas peddle. The hesitation at start up only happens at start up but the slight bog down stays with the car at a stop. Anyone got any areas I need to focus on to cure this issue?

 

What do you mean by "loads up" and "hesitates"? A full second of cranking before catching is quite good. Mini doesn't view it as a problem until cranking is in excess of 5 seconds and most people won't even pay it any attention if less that 2 or 3.

The bog down when pushing the peddle could be spark or gas related. Lots of options from spark plugs, wires, coil to fuel pump. How many miles on the car? What color is it?

 

I have solved many strange starting/idling/hesitation/driveability issues simply by doing an ECU reset (search this site for a DIY).
Try that first. If it doesn't work scan for trouble codes, read your fuel trim values and report back.

 

Had a similar issue to this on a car I had before I bought my mini, I had my thermostat replaced which helped a bit but it still had an iffy idle, then I had the valve cover gaskets replaced and that seemed to really help!

 

The hesitating I have is right after it cranks. It cranks right up, and the first 1/2-3/4 sec it makes a quick sound that is either hesitating, or loading up. Resetting the ecu did nothing. Scan shows a Cylinder 2 DTC misfire

 

One thought....
Long cranking, followed by a "lumpy" start can point to a "drippy" injector, resulting in both loss of prime (the reason for the long crank) AND the rumbly start as it runs excessively rich in that one cylinder as the liquid fuel is burnt off....and the hesition from a non moving start would indicate a poor spray pattern, likely from the same injector....
I'd run a couple bottles of injector cleaner through it, see if it smooths out...sometimes some gunk can buildup....used injectors pop up pretty cheap too....

 

I had the "long crank" and "rough run" at start up - turned out to be a bad injector. Dealer replaced and problem solved.