I am considering changing out the stock 2.77 gears in my 87 Trans Am, looking to go to 3.23 or 3.42. What is involved with this change and will it require any modifications or reprogramming to the transmission itself? Also, how much of a benefit would an aluminum or carbon fiber driveshaft be to a basically stock setup? Would it be worth the cost?

No, a new driveshaft is not worth the cost, although some people say it's worth it to get rid of the vibration.

First off, you need to know if you have 2.77 or 2.73. One is for a 9 bolt and one's for a 10 bolt. Second, you need to take off the brakes so you can push the axles in. There's a big rod in the middle of the carrier in the axle. Take that out and push the axle in. There's a c-clip holding the axle on, take that out on both sides. Unbolt the carrier and take it out. Reverse that for install.

If that's wrong go ahead and correct me.

Ugh! What a subject,,,I will try to make this as short as possible for you. The Aussie 9 bolt rear has bolt in axles, no C-clips,,,kinda like a 9 inch ford only smaller.
First you have to determine what you have,,,,9bolt or 10 bolt
the 9 bolt is expensive to do the gear change in since parts are hard to get,,,I have had no luck contacting the fellow everyone mentions on here,,,,that sells 9 bolt parts,,,,in any event,,,if you have a 9 bolt and 2.77 gears,,,you will need a new posi carrier for the 3.27 or 3.45 gears,,,,plus a setup kit,,,if you can find those parts and have the tools to set it up correctly then your set.
If you have the 10 bolt,,,things get a little easier parts wise and a little less expensive....if your 10 bolt has 2.73 gears,,,you can jump to a 3.08 gear before you have to change the posi carrier,,,from there 3.23 and up require a different carrier,,,but the carrier is still cheaper than the 9 bolt version. Again a setup kit for the new gears and the proper tools to set it up and you are on your way.
I can tell you what I am in the middle of doing,,,I have an 88 Iroc with the dreaded 9 bolt....It's coming out,,,that simple. The easiest thing to do is pick up a 10 bolt out of any third gen,,,,,they are a dime a dozen,,,I found mine for 10 bucks complete I am buying a new Eaton series posi, new 3.23 gears, setup kit,,,new axles etc.....then removing the 9 bolt and installing the much more parts friendly 10 bolt,,,,no more fuss. Larry.