I've been checking in a handgun on trips for many years.
Here are some key issues not covered above.
Ammunition is not difficult. I generally carry one box of whatever load I want in my carry gun. (Because it might not be available where you are going, you might arrive after dark, plus, driving around an unfamiliar town trying to find a box of specific rounds is a PITA.)
If you carry ammunition in your checked luggage, it must be in the orignal box. Don't, for example, pack it in one of those plastic after market boxes.
I do not pack the ammunition inside the hardshell case, unless the hardshell case is designed for that. There is a hardshell case made by Winchester that has an unpadded compartment inside the case for a box of ammunition. (On one rough ten hour flight the contents of the hardshell case moved so that the pistol was rubbing against the cardboard box, and I was not happy with the results.) Here is the case:
http://www.exploreproducts.com/handguncasewgs7702.htm
There are limits to how much ammunition you can pack with you. For example, Delta's website says "...We do allow small arms ammunition, in quantities not exceeding 11 lbs. (5 kg) per person"
When you go up to the check in counter, tell the agent you need to sign an orange tag. You sign the tag and place it inside your checked luggage, but NOT inside the hardshell pistol case.
One reason that I don't use the plastic case your pistol came in is because try as you may, you might not be able to keep the pistol on your person on every occasion on every trip. It may need to be left locked in a rental car trunk, or possibly locked in a hotel room, inside your suitcase. It represents a potential liability, so I use a trigger lock and lock the pistol inside a robust aluminum pistol case I bought at Cabelas, or in the Winchester aluminum hardshell case mentioned above.
Also, as brickeyee mentioned above, not every rental car HAS a trunk. I carry a cable padlock to be able to lock the hardshell case to something rigid inside the rental car (under the seat, etc.) in the event that my travel plans suddenly change. Travel plans have been known to not work out as you intended...
Some airports - for example, Denver International Airport - have a rather time consuming requirement. After you check in and declare the firearm, sign the orange tag, and lock it inside your hardshell suitcase, the ticket agent calls over a security person who places your suitcase on a luggage dolly and rolls it over to a TSA office. At the TSA office they scan your suitcase while you watch, and affix a tag to it indicating that it has been pre-scanned. Then you take it back to the ticket agent, and your bag proceeds back to the luggage area on the standard conveyor belt. It easily adds 25-30 minutes to your processing time.
If you know about it in advance, you can allow for it. But if you don't, it can make a tight schedule much less enjoyable. Therefore, I generally allow at least an extra half hour - or a total of 2.5 hours before take-off - just because airports are unpleasant under the best of circumstances, but having a gun in your bag can complicate your anxiety significantly. Give yourself lots of time. You can always grab a coffee and a paper and relax if you have extra time after getting in through security.
Finally, don't take your prize whatever that was your Dad's, that you won, that you spent $3K on, whatever. I've had two suitcases disappear and simply never be found. The airline carrier pays you off with pennies on the dollar, issues a letter of apology, and ignores any ranting you do afterwards. Good luck with trying to sue an airlines. [Edit: the two suitcases were lost on overseas flights and did not have firearms in them. I have not lost 2 handguns. But it happens.]
So, carry a work gun, something you don't mind losing if it happens. Have insurance on it, photographs of it, receipts and serial numbers, etc.
Happy Trails!
Doc