That's an EXCELLENT question! And it's something that we don't cover very thoroughly on the site (not yet, anyway).
There are certain undershirts that are better than others.
There's a company called Advantage Wear (www.advantagewear.com) that specializes in products for excessive sweating.
You can find underarm sweat-shields (which I highly recommend), and also shirts with the shields already sewn in. Their clothes are a little more expensive than regular clothes, but I've used them with good results.
Another company that is dedicated to making good excessive sweating undershirts is called Kleinert's Shields (www.kleinertsshields.com). They have a huge variety of sweat-guards, and clothing specially designed for excessive sweaters.
The last thing I would recommend to you Joshua, with regard to undershirts is to try wearing more than one. If it's not too warm, stacking a few undershirts can help a lot.
Speaking of Stacking, have you been following "Mechanic Method?"