To some extent it depends on when you plan to arrive and how you will get under way. I like to ride right out of the airport without getting a room, waiting around for a bike shop to be open, or looking for where ever the bike was shipped to and so on. So I like to fly with the bike for most trips.
I usually fly Southwest so taking the bike is reasonable enough. Last time I checked SW charged $75 for the bike but the other bag was free. I think Delta is $150 for the bike and they charge for the second bag too.
I have never had any damage to the bike beyond possibly a little normal wear and tear that I didn't notice. Properly packed the risk of serious damage is quite low.
If you will get a room directly from the airport using public transportation anyway then shipping the bike using bikeflights.com or ship bikes.com may be the way to go. Staying with a warmshowers.org host and shipping the bike to them can also work.
On the way home I like to just drop the bike at a bike shop and have them pack and ship it. I am usually happy to be shed of the hassle of packing and shipping a bike in a strange town. They usually charge $40-60 for the packing and $40-60 for the shipping. It has always come out at about $100 for the whole deal.
One thing to not do is go to a UPS store and have them ship it the rates are always jacked up when you walk up to the counter. In fact I have typically been charged more for the shipping that the total for packing and shipping via UPS at a bike shop. UPS stores, Postman Plus type stores, and probably FedEx are sure to gouge you for an extra $50 in my experience. I say probably on FedEx, because I have not used a FedEx store.
Last edited by staehpj1; 03-15-17 at 06:38 AM.