OK, first of all, the tourist visa is ridiculously easy. All you have to do is go to the Australian Consulate's website, fill out a short form, and pay $20 (I believe that's $20 Australian, wich may be more like $22 US; I haven't checked the exchange rates this month). You tell them your passport number but they don't need to actually handle your passport. Everything is done electronically. When you arrive, they'll scan your passport and your visa will show up.
Now, about hostels - most Aussie hostels and backpacker hotels have single or twin rooms as well as the 4- and 8-bunk dorms. Check em out online. They typically cost less than a room at a regular hotel, and you'll meet a lot more people than you would at a regular hotel. And, they're popular with people of all ages. I stayed in single rooms at hostels exclusively on my trip in July/August, met a lot of great people, and I typically was not the oldest one there.
If you just can't bring yourself to stay at a hostel, many towns have "pub accommodations" - cheap rooms above a corner pub. They're also typically cheaper than a regular hotel, and can be a fun stay. I used to love staying at the Lord Nelson in Sydney - but, unfortunately, they remodeled and their rates skyrocketed a few years back.
Now, for what to do in January - it's mid summer down there of course, and the height of the tourist season. But it's also the wet season up north, so I wouldn't go up to Darwin or Cairns - terribly hot, humid and wet, and always the threat of cyclones. Also poison jellyfish at the beaches. And the diving isn't so good that time of year; the rainstorms muddy up the water.
At Ayers Rock it will be ungodly hot (but dry) and swarming with flies. Don't go. It's just a big red rock anyway.
Sydney and Melbourne are great that time of year. If you plan on renting a car, a drive around the coast between these two cities is nice - but you have to like the outdoors; there are only very small towns in between. Near Melbourne, Wilson's Promontory National Park is a beautiful place with lots of wildlife roaming around (kangaroos, emus, etc). Just beyond Melbourne is the Great Ocean Road, Australia's version of Big Sur.
If you have time, and if you're a little adventurous and outdoors inclined, you may want to hop over to Tasmania. It's a short flight or an overnight ferry boat from Melbourne. But be forewarned that the weather is very unpredictable. To the south there's nothing but open water all the way to Antarctica, and to the west, the next stop is Argentina, so storms can blow in from anywhere at any time.
More specific questions? Feel free to send me an e-mail. I worked over there for a while, and have been back many times as a tourist.