Note that axles are usually ~10mm longer than the hub spacing. So, if you have a typical touring/MTB hub that's 135mm, you'll usually go with a 145mm axle. You can go a bit shorter, but not much. You want a little bit extra on each side to fit into the dropouts for alignment.

Again, the cones and nuts are the parts that may be specific to the way your hub's bearing races are made. Also, before taking it apart, note which spacers go on which side. It can be difficult to get things lined up again afterwards if you don't.

If the washers from your old axle have orientation tabs on them, as MSN Tourist notes, either get new washers or simply file down/off the tabs. They don't really serve much of a purpose.