Instead of what oil...maybe a better question should be why first. Gear diffs can be a very useful tuning option for modern day touring cars. Now that companies have figured out how to get them to be light enough you can now pretty much run them in both the front and rear of the car exclusively. First lets start with the rear diff.
Rear Gear Diffs:
It is important to understand how the gear diff is constructed so that you can make a good plan on how the diff should be filled. Some gear diffs have a small cavity and others much bigger. As reference I will use the Xray gear diff as an example as I think that is about in the middle volume wise. Most drivers run 1K to 2K in the rear gear diff for stock and super stock racing because it helps the car carry speed through the corner, but limits the amount of forward grip. Mod drivers tend to go thicker to keep the rear end more planted and provide additional forward traction. A good place to start in mod is 2k-3k. Based on traction or tires you can continue to go up in weight until the car feels balanced. Temperature will have a big impact...so if you are running in hot weather it is best to go thicker to accommodate the thinning effect you will get. it is not unusual to run 5k or 7k in mod to help calm down the rear of the car.
Front Gear Diffs:
The thinner the oil the more turn in you will get while losing traction out of the corner. 100K in the front will give really good turn in and mid corner steering but will diff out when you get back on power coming out of the corner. Putty sometimes will feel too much like a spool. My recommendation is always to start with Gravity 2.5million CST in the front. This is almost always the right place to start because it provides more turn in then a spool. Coming out of the corner it wont have as much rip as a spool, but it provides great forward traction. If you need more turn in go to 1million cst in the front but understand that what you get coming in you will lose going out.
Some gear diffs you can run with 4 satellite gears or two. Two gears will make the oil feel thinner, so in some cases you might want to adjust for that, but this can also be used to adjust feel without having to go thinner on the oil. In the Awesomatix a two gear diff feels much lighter on the track then a four gear diff with the same oil.
Last point on gear diffs...they require some driving adjustment to get used to them. They will allow you to drive a bit more aggressively coming out of the corner, but going in might need a little bit more finesse. Since a spool will push while initiating the gear diff might oversteer, so be ready to adjust for that. Keep in mind that team drivers tend to have some insight from the designers that force them to pick a certain diff oil. So I would check some setup sheets or PM some people to find out where to start with your specific car.
Good luck and Hope this helps!