In addition to getting a trigger with a lighter pull weight, I also have the option of switching from a single stage to a two stage trigger. What would the pros and cons of doing so be?
What the 2-stage gains you is having a perceived lighter trigger weight even though it is the same overall. I forget the technical term for it, but the first stage takes up the "slack" or travel in the trigger up to where you reach the second stage. From there it acts just like a single stage. So the potential benefit is if you can set the first stage high enough, you need just a little bit more squeeze to fire the trigger.
For example, in your other question we assumed you had a 7-8 pound pull. With a single stage you keep squeezing until you reach that 8 pounds and BANG. With a 2-stage, you could set the first stage to say 6 or 7 pounds (depends on the springs how much adjustment you have). So now you pull back 7 pounds and you reach a stop. At that point you know it's just another pound until BANG.
There isn't really much against having one other than cost and being slightly more complex. For hunting I wouldn't bother, but it would give you something to play with. :)
I can't say for sure on rifles how well a 2-stage trigger works, but I like it for my revolver.
Personal preference for me would be to go to a lighter trigger on a rifle though. This is primarily to reduce the finger squeeze pressure you're exerting to complete the first stage. I feel like that could potentially throw off your shot.
However, for me and my revolver, it is pretty nice. It doesn't have safeties and no external hammer. So, I can pull partway, better line up my shot, then complete it for a more accurate shot.
When it comes to long guns, I would prefer a lighter trigger since I would have those external safeties. As in, when I put my finger on the trigger, I mean to shoot very soon. I would rather not have an intermediate step in between that process.
It really comes down to personal preference and which option you work better with. Try and compare both options and find what works best for you personally.