We've moved a few times. We've used both http://www.advanceremovals.com.au/
(Geoff was fantastic) and Grace Removals and would recommend them both.
The last move we had was by far the easiest, that's because we had packers, they organised house, window and carpet cleaning.
I can only suggest to seriously cull (as you have already pointed out).
If you do pack, get proper boxes/paper.
Storage isn't cheap, it really depends on how long you want to store your belongings for as to what the quotation will be like.
Each time we have moved, we had already lined up a place to move to which does help (usually vast research via the internet and then a reconnaissance few days around the places we want to live).