I think you'll find that the anticipation is the worst part.  Once you are actually on the plane I believe you'll find it fun!  People who fly all the time have lost that zest, but virgin flyers should view it as an exciting treat.

Allow yourself plenty of time when you check in.  Nothng adds stress and fear more than thinking you're gonna miss your flight.  Remove EVERYTHING from your purse and/or carry-on that might be suspect (small scissors, pocket knife on keyring, anything metal you don't need, etc.)  Carry-on a good book.  Have gum in your purse if you need to clear your ears. Try to get a window seat so you can look out...you'll be impressed at what you see from your perspective.  Order a cocktail as soon as possible.  If there's no movie, listen to music...that's quite calming.  If you're changing planes prior to your destination, learn all you can about the airport and gates before you get there...and be sure you have plenty of transfer time so you won't be running and stressing  if your first flight is late or delayed.  If at all possible, travel light and carry your bag on-board.  Then you won't have the added worry about losing your bag before you get there, or having to wait to claim it when you do.  Read the on-flight gift magazine...they have really cool, unusal items you'll never find anywhere else.

And always remember...flying is much safer than driving.  You'll be perfectly fine and have a good time too.

-frolix