Do something happily and you'll be cheerful and lighthearted. It can lift your spirits just to hear your friend whistling happily as she walks down the sidewalk with a smile on her face.
If you're happy to do something, you'll do it happily. Most of us will happily finish that last piece of cake or happily cuddle a tiny puppy. A second meaning of this adverb is "fortunately," so you could say, "Happily, I shut the front door before your cat got out!" This definition dates from the late 14th century — earlier, people used happily to mean "accidentally." And of course the very happiest stories end with "they lived happily ever after."