I had installed last year a new coleman furnace and central air conditioner. My electric bill doubles when either the furnace or AC is on. Is it possible that the wire size to the blower is too small. The company that installed it used 14-2 wire.
No, that isn't causing your $ increase.
The wire size is almost certainly irrelevant to the size of your bill. Heat and AC use lots of power, so cost lots of $$.
Your question is similar to saying "my water bill is really high when I fill my pool or irrigate my yard, should I use a larger hose?" If you think about it in those terms you will see why larger wire wouldn't make a noticable difference.