I came oh so very close to putting my bike on a plane with me, but ultimately did not.
Why?
Chalk it up to price. I do know a few people who have done it and can summarize their experience.
You will pay an absolute minimum of $50 to ship your bike, almost always per direction of travel (meaning minimum $100 for a roundtrip). Although I've heard British Airways doesn't charge for bicycles, I am guessing they don't run a Seattle-LA route
Catch is this - the prices to ship the bike with you vary, from a minimum of $50 as mentioned above to about $200 - again these are one-way prices. Plus the airline with the cheapest bike shipping price like say Alaska/Horizon may not have the best prices on the plane ticket itself for you. So you kind of have to factor in total price if you are on a budget.
Usually you have to ship the bike w/o pedals, bars turned sideways, and in a box. You can go
online and buy bicycle boxes for shipping if you can't find them locally. Yeap, one more cost to factor in your travel.
More info
here and
here:
I would check fares/routes/schedules then verify with the airlines themselves of their current policies since they are apt to change w/o notice. Also remember that the bike counts as a piece of luggage so you may pay more for your regular baggage too, since almost all airlines now charge for bags...
Overall the feedback I've gotten from friends who traveled with bikes has been positive: bike made it OK, some extra hassle in prepping/planning with the airline, and of course more money. For a short trip it probably is easier/cheaper to bring your saddle/pedals and rent a bike but if you are going to be there a while, then your own bike seems like a more economical choice.
Happy travels