Mattress cover packs that you can pick up from your local storage facility or Bunnings are great if you are storing your items however when moving mattresses, having these covers can make it very hard to move.
Mattress covers tend to be very lose, so they can fit a variety of sizes of mattress but when moving it makes it hard to grip onto the mattress and if access is restrictive when moving this can make it hard to manoeuvre your mattress.
A reputable removalist will always shrink wrap your mattress which provides greater protection when moving your mattress and makes it a lot easier to manoeuvre.
Shrink wrap is available from sotrage facilities and your local Bunnings as well.