Re: BRAND NEW TO SHOOTING
First check with the airlines you will be flying. Some have some weird rules, ie no more than 1 gun per case.
You must transport them with the airlines in hardcase that is lockable. Easiest and cheapest is put them in padded softcase and then in cheap plastic and lockable golf bag case. You can buy them at Walmart or Kmart often under $50 and will hold 3-4 guns and does not look like guncase.
As for cheap gun to start the 22 long rifle is the cheapest and fun to shoot. Think you will find the heavy barrel types more accurate, such as Ruger 10/22 target, Rem 597 in bull barrel etc. Can be had for normally around $300. Most reputable gun stores will have. Advice is to go look, feel and see what you like.
The ruger can easily be modified with stocks, barrels, tuned triggers, etc that make it fun to modify and shoot, but the stock heavy barrel with a little trigger tune is more than acceptable.
after that a good 223 or 308 is easy to shoot, low recoil, accurate and relatively cheap to shoot. One of the best deals is the savage heavy barrels such as the BVSS for around $400 used or $550 new.
Good luck and good shootin