The cheapest new "real" bike you will find is probably the Giant Boulder 4, this is a MTB-style bike with a low step-over height (a consideration with a child seat, and so low that the W variant isn't really worth it). A quick look at BikeExchange suggests the going rate is $350. Don't get fussed about 2011 versus 2012 models, they're the same in essentials. Like all basic bikes, over time you'll want to replace the basic rubber tyres with something more reliable (eg: Vittoria Randonneur Pro Touring, about $80 the pair).
If that's beyond you then ring your state's bike organisation (eg, BikeSA here in Adelaide). There are old blokes who like to keep their hands busy by grabbing bikes from the hard pickup, fixing them up, and selling them at cost to worthy people. The bike organisation will know who they are (or will know who knows who they are).
There are also for-profit people who re-sell old bikes. Some suburban bike shops (eg Prospect in Adelaide) and they're often at markets (eg, Sunday's Port Adelaide Market). Some are dodgy (stolen bikes have to end up somewhere) but those tend to be the more "pawn shop" or "general second hand" stores rather than bike-specific places.
Hit Internet sites like BikeExchange hard, sometimes people drop their old bike there and aren't really fussed about the price so much as clearing out their shed. These people really want to minimise their hassle, so you can anticipate needing to collect the bike from them. Sizing can be hard with Internet sites, as the best way to size a bike is to step over it and to ride it. That's where a local bike association can be a big help, you can step over and sit on their hire bikes and get a good idea what your size is for differing sorts of bikes, whereas if you try that in a shop the staff may well get upset when they realise you aren't going to buy from them. If you ring the bike association in advance you'll probably find that you'll get one-on-one help when you go in.
Make your choice based on: (1) fit, you'll want to be comfortable and in control; (2) not exotic, you want to keep this thing on the road cheaply (so take the bike with solid forks over the bike with suspension forks, something from the 2000s is better than the 1980s in terms of availability of parts, and avoid anything earlier than 1980); (3) weight, a light bike makes a huge difference. Make sure it has lugs for a rack (which is how the child seat will mount) and lugs for mudguards. Make sure that any future mudguards will fit under the brakes; also suspension and mudguards often don't mix.
Don't be too fussed about the type of bike. A MTB would be good because it is lower, the wheels are wider and both of those help with stability with the child seat (you can always rip the knobbly tyres off the MTB and put on 26in slicks). But a hybrid flat bar bike with bigger wheels will go faster and more easily. So rather than decide in advance look at both categories so you can get the best bike available for the low price you need. Don't buy a 1980s "10 speed" as these were crap. Don't buy a "racing bike" or any other bike with thin (eg 21mm) wheels: the weight of the child in the back will upset the balance of those bikes and you'll go over for sure whenever there's any hint of a loss of grip. Hopefully that sort of bike won't have the pannier lugs anyway.
Note that child seats often come with their own pannier rack and another rack won't be able to fasten the seat. So buy the seat first.
Front baskets are a bit tricky, since it's easy for them to foul the cables or front brakes. So take your bike along when buying the basket to check the fit. Many people find a small backpack to be more practical than a basket. You've got to check the fit of backpacks too -- you don't want to whack junior in the face.
Locks and lights can be more expensive than you expect. Since you are at uni, ask them about secure bike storage places -- they'll often need to add access to the bike cage to your student card. Then get the heaviest smallest U lock you can afford (and Internet sites like Wiggle
, Cell, etc can make that money go further). Leave it locked to the post in the bike cage so you don't have to drag it in each day. A lot of people will buy a $10 cable lock to fasten the wheels to the bike, that might be overkill but the thing about commuting is that you need all of your bike to be there at the end of the day.
Whilst you are on that Internet site, buy the lights as well. At your budget you are looking for a good "be seen" light rather than something which will light up the road -- a 1W LED with a flashing pattern will do the job. You want rechargable batteries for the front light: the only question is if the light has those built in or if you buy rechargeable AAs and a charger from the supermarket. Also go down to your local "safety" store and buy two metres of reflective tape. Cut it up and stick it to that child seat until it glows from the back and sides.
Clothing-wise, for a short commute like yours the most valuable thing is a lightweight jacket (very much like a spray jacket, but cycling specific so the water doesn't wet your back by creeping up from the road spray). These are enough to cut out the wind and stop the rain from freezing you (you'll get damp but you shouldn't get wringing wet). You'll see these on sale at times (Katmandhu is having a sale now). Junior isn't doing the work, so they want something warm in the rain, like a parka under a plastic footy poncho. Make sure the water doesn't pool in their seat or wet their legs, you might need to peg the poncho to the outside of the seat to stop that happening. If your commute was longer, then cycling knicks would be a good buy, and even on your short ride if you get a lot of chaffing in sensitive areas you might want to ask for a pair for your birthday.
Finally, interesting a cycling friend is a good way to make your money go further. You can borrow tools rather than buy them and they can show you the basic maintenance (which is simply to keep the bike clean and the chain lubed). Even a beginning cyclist can steer you away from a crap bike, and an experienced cyclist will be able to find the gem you want in a sea of second hand bikes. All old cyclists are good at tracking down a bargain, but you might need to lower their sights so that they pick up your entire package for less then they might spend for a single wheel.