Piece Luggage Set

What kind of bag/luggage is suppose to be used for traveling on airplane and on cruises?
Is it acceptable and/or appropriate to use a duffel bag instead of an actual piece of luggage for traveling through airports and taking on cruises? I just bought a luggage set for my husband and I, and it comes with a duffel. With the pieces it came with I am guessing one of us will have to use the duffel, so we can bring the things we need, but I was wondering if this will be frowned upon or not. This will be my first time on an airplane and going on a cruise, so I don’t know what is acceptable and appropriate. Also if anyone can tell me if bringing hair dryers and styling tools is allowed or not I would appreciate it!
For luggages on a plane your concern would be whether it is sturdy enough to withstand handling, it will not tear easily, that it is secure enough (can be locked), esp if it is to be checked in/not hand carried in to the airplane. People travel with all forms of bags, not a problem. I bring hair dryers/straighteners etc., not a problem – you will want to wrap it in something (I use a towel) to avoid damage. For hand carry into the airplane there is a size and weight limit – you will need to check the limit at the airport.
Snakeskin Print 7-piece Luggage Set by Travel Concepts