Having just done this to the Czech Republic with an experienced flier, I can give you what I learned on this trip for International travel. (some of this will not pertain to US flights only)
*We insured all guns and scopes we traveled with. We added them to our homeowners policy. As with any insurance, you hope you never need it, but have it just in case!
*If you can, book all travel with just a single airline. For example, we flew Delta the entire way.... we had to get up early because if you took the later flight it was a code share with another airline.... transferring to another airline just means more chances your bag/rifle could have issues making the flight. Keep it simple, even for local flights (not international)
*Be prepared for extra baggage fees.
*Have a very good hard case with foam cut exactly for your gun, scope and gear.
*Ammo/pellets must be separate. We put them in with our clothes.
*Have good locks for every lockable part of the case. If it has four spots then you must have four locks. TSA requires this.
*We used air tags. A must for peace of mind and yes, we used them to be sure the guns were making the flight and arrived with us.
*Guns must be completely emptied of air and the valve removed from the bottle so it can be inspected.
*If flying International, Take your guns to the CBP office about a month ahead and get the Customs paperwork (don't remember the name) that records the serial numbers of the guns and scopes you are taking. They inspect the guns and record them so you can return with them.
*Know the countries laws and what permits you need, if any. For example the Czech Republic did not need a permit for the Air Rifle, but it did for rimfire.
*Have an invitation letter from the club that you are going to shoot with. This was looked at every time the guns were inspected. Have several copies of it with you. We presented this when asked for a "permit" It was at this time we say it's an air rifle and we have an invitation letter.
*ALLOW extra time to clear Customs and Inspection points. Our layovers were 3 hours at JFK both going and returning and we needed it!
*Know your air availability on the other end with the club you are going to. We did not fly with a Buddy bottle, but needed one. We were able to borrow on the other end, but don't forget to find out about air.
*Make sure you have a sight in day and rely on your previous knowledge of your tune. Even if you have an issue, remember.....This is fun, learn for the next time you go. Do the best that you can do and give it your all, but.....this is fun! Don't stress it.
I am new to this, but hoping this helps you to be able to travel and have fun at further distances!