Moving guns around the states and into Canada is not a problem. I would suspect NRA could give you sound advice and also go on Google and do a search there. I have travelled with hunting rifles and shotguns on numerous occasions since 9/11 as I have hunted ducks and geese in Wisconsin, South Dakota, Nebraska; big game in Canada, Alaska and Wyoming, and doves ducks, pigeons and perdiz in Argentina. You need an airline approved lockable case, I use combination locks and then I use small zip ties to reconfirm the locks were not tampered with. Someone earlier mentioned 11 lbs is the limit on ammunition and must travel separately. That is the limit you can declare. I have seen higher weights transported but 11 lbs is a legal limit. Interestingly since ammo is an explosive that is the one item the airline Captain, since he is responsible for the safety of the plane can actually refuse to haul ammo. If concerned just plan on buying new ammo on arrival as ammo is heavy and really counts against your weight limits. I buy fresh ammo at my destination. Remember to research ahead on where you can get ammo especially if you use special tungsten loads etc. Airlines will haul ammo but the airline captain actually has the right to refuse to haul ammunition. Remember guns and ammo must travel separately. Get to the airport an hour earlier than normal and just expect a hard time as not all clerks are used to handling weapons. They are all taught how to handle the guns but they forget the procedures and have to ask supervisors etc. al
I have been stopped twice for not having cleared my carryon bags of shells. On one occasion I had a 30-06 round lodge against a plastic seam and I took that bag through about 4 X-ray exams on several various trips before an alert TSA agent spotted the round logged in the small hunting backpack parallel to the seam. On another time I had several shotgun shells in the end pocket of a blind bag that I never sufficiently cleared of shells. In both cases I was allowed to continue on after I forfeited the ammunition as they were honest mistakes. Now if my name were Mohammed I am sure the story would have been different.