Hi eatchow, welcome to the forum. Before get started I will say that I am by no means an authority on this subject but having done something similar, can offer some advice. Since no one else has answered, I thought I'd jump in.
Firstly, your previous post says you would like to build a"period correct"car. Abarth made many variations of Fiat's 600 model and for this reason I think you should do some research before getting stuck into the project. There are many books available that will cover these variations, most sold on the internet (ebay etc). For example, you could make an 850TC replica (and there were different versions of these made)or a 1000TC replica. The major difference between these models is the size of the engine but for period correctness there are the little things like radiators, badges, suspension, brakes and lots of other small things I can't think of right now. It will help you if you have a reference point to focus on. If not, you may end up with a "bitsa"car like mine! Fun to drive, works better than the original but not "correct".
I would recommend joining your local Fiat Car Club as the knowledge of old Fiat guys is staggering. They will know what year model had this bit or that bit. Many of them have rare parts stashed away, or know where you might find them.
In regards to your actual question; You need to find some parts from an 850 model. This would be the cheapest way to convert to disc brake front. The later versions of the 850 had a disc brake front end which will fit the 600 exactly (the 600 rim might not clear the caliper but you will probably have better rims, yes?). Also, you will find that alot of the 850 will fit the 600, with varying degrees of correctness.
The period option for suspension was to lower the front of the car by reversing the lower leaf of the front spring. You can buy these complete but the local suspesion specialist should be able to do this for you. For that reason you would want the tire/rim situation sorted first. With that done you can tell him exactly how low/where you want the car to sit. You may look at fittng a front sway bar at this time. Again, the 850 came with this as standard but there are no mounting points on the 600 body so some fabrication will be required. Original Abarth's had a small strengthening bar fitted to the top of spring that attached to the underside of the body but I don't know how effective this was.
For the rear of the car, lowering the ride height and some better shocks. Beware of stressing the original stub axles with more power,more grip from wider tyres and lower ride height. A broken axle may result in rolling the car.
It might be a good idea to check on the class regulations of the racing you intend to do. You may not be able to race in a particular class if your car is too heavily modified and never homologated that way.
The forums are a great way to get information also. There are a few people who frequent them with some great advice. Guy, on this forum. Jeff Sitch (aka viotti600) seems to have an encylopedic knowledge of early Abarth's and I think is in the U.S. Look out for him on this forum. Also one can't forget Paul from Scuderia Topolino. He runs a go fast business in the U.S and does some very serious racing himself. Check out his own car for what must be the most advanced Fiat Abarth based 600 in the world! I mean this thing wouldn't look out of place on an F1 grid...
Good luck with your project,
Toby.