Wouldnt worry that much - from reading your previous post, you are coming over with DHCS and they are bending over backwards to help with relocations. January isnt a great time, no, but February is possibly worse as the uni students are on their way in. You are a different demographic from the uni student brigade and family homes, especially if you are up for a slightly higher rental, are going to be in marginally less demand. As a public service family, too, you are going to be a better risk for the estate agents who do the letting. You would be wise to get here so the kids could start school in February - if you get in before the census and you want to try and negotiate an out of area placement you stand a better chance.
Canberra is a nice place really despite the Sydney/Melbourne dissers - the closer in to the centre you get the older the homes and the bigger the blocks but the more expensive they are (except for the very inner city which is now apartment living). The new builds are IMHO soul-less places and though every man and his dog is now living in Gungahlin it really chills me. I am an inner North girl and you could do worse for both schools and homes than to look at Reid, Campbell, Ainslie, Downer, Watson, Hackett or even across the lake in Yarralumla, Griffith, Forrest (how rich are you?) Garran, Hughes.
As for bringing things - well, you are likely to be able to replace pretty much anything but it will cost you - although in January there will be the post Christmas sales so you can pick up good bargains and now that we have a Factory Outlet shop in Fyshwick the world is your oyster.
Assume you keep an eye on
Real Estate & Homes for Sale, Rent and Share | allhomes just to see what is on offer and what sort of houses there are in each area.
You will be fine I am sure!