Use 'exhilarate' to describe intense positive feelings, more than just 'happy'. The roller coaster ride exhilarated the children, leaving them breathless with joy.
Pro Tip 2/3
Often Personal
'Exhilarate' is generally used when someone's feelings are affected personally. She was exhilarated by the surprise birthday party her friends organized.
Pro Tip 3/3
Not Subtle
The effect of 'exhilarate' is strong and obvious, not a quiet or modest feeling. Winning the championship exhilarated the whole team; they couldn't stop smiling.