First is the day or weekend bag it can be big enough to carry everything you need for a long weekend plus all your clothing etc or smaller overnight style bag just to carry your clothes and a few extras like reels flyline etc. Again it comes down to that personal choice. When traveling on longer trips by air you will need a good size travel duffel. My favourite has four compartment the two end compartments have a good strong mesh to aerate the pocket so good for waders, boots and dirty laundry. The middle section is larger and is used for your clothes etc. I like to pack my reels spare flyline and other valubles in amongst my clothes to protect them. The bottom of the bag is one large storage area for your fly rods landing net and anything else you can fit in.
What ever bag you choose be sure that it is long enough to fit your fly rods. Double lockable zippers for security and wheels and extendable handle for easy transport, remember it's going to weigh about 20kg or more. Make sure that the bag you buy is super strong. The amount of damage I have seen to bags over my many years of hosting overseas trips is unbelievable so choose carefully.
For a small fee of about 5 or 6 dollars you can have your bag wraped in pallet wrap this not only protects the outer surface of the bag but stops baggage handlers from pulling your bag out from a pile of 20 other bags sitting on top of yours by the handle. This is how bags get destroyed. Be sure to organise your bag so you know where everything is, that way you don't have to empty the whole bag just to find you line cleaner.