Skirt boards are one of the hardest things in my opinion to do. I can rip apart a bathroom and have it put back together in 2-3 days and it might take me the same amount of time to make a couple of long stair skirts. All of your stairs wont be the exact same measurements, walls are not perfectly flat... One of the things that I source out now for sure.
The short answer is it depends. If the staircase will be carpeted or if there is no overhang on the treads then the answer is tread first. Basically this is faster because your treads don't have to be perfect. Also your skirt doesn't have to be perfect.
If your stairs will be showing the treads or if there is an overhang I would suggest installing the skirt and then the treads. This will give you a cleaner look. Also good luck notching out the skirt to match all of the overhangs on the tread if you decided to do it last.
There are 2 ways to do this, the first way is in essence just cutting the wall at the top of the skirt board and capping it off and setting the handrail on top of the cap that is finishing off the cut wall, sounds a bit crude, but essentially that describes what you will have.
The up side to this is you will not need to do a floor repair, depending on where your wall ends where the newel will be.
The other way to do it is by removing the wall entirely, repairing the floor, removing the treads and risers, and set longer ones with return nosings on either side. and re-support the stairs with a new walls under the existing carriages.
This is the most work, but will give the best result in my opinion.
Jan. 2, 2014 edit
Here is the work I did in 1989 that shows what I did. This one shows the best example of the effect I was trying to explain. I do not have a shot that shows how the handrail goes past the ceiling, but with a little imagination you could picture what the rail would have to do if the stairs were much wider at the bottom.
Edit #2
I did a little calculating to help, if it may
Best Answer
There are special screws for squeaky wood floors. You drill a small pilot hole, drive in the special screw and the head of it breaks off. The screw pulls the offending pieces together eliminating the squeak. You will want to try to screw into the stringers for the best result. With the head broken off a small hole remains that can be filled. If you have large gaps between the treads and the risers you may want to get some wood glue in the gap prior to installing the screws.