Drill a 1/4 hole, then a 1/2, and then the 1"
I'll disagree with this, drills tend to not like to go down a hole that is even remotely close to its own size. As someone else said they get grippy/grabby, which just adds to the problems. My general rule of thumb is a drill doesn't go down a hole that is more than 1/3 its diameter.
Also there is a lot of shaking going on
If your going from 3/4 to 1" that will happen.
On a 1" drill, I would measure the web, and then just drill a pilot slightly larger, and then stuff the one inch in.
Do I increase drill sizes until I reach the desired size? Do I slow down the drill speed as the bit gets larger?
This is where you go back to the basics. Surface feet. I've found on this site and from personal expierience that a lot of beginners think in RPMs, I guess its natural. However thinking in RPMs can bite you in the ass real quick.
If your thinking I cut 'X' material at 1200 rpms and 'Y' material at 600 rpms, your going to eat a lot of mills/drills/inserts. If its stuck in your head 'X' material at 80sfm with HSS and 'Y' material at 600sfm with carbide, then your pretty far along the learning curve, and will save yourself a lot of headaches in the future.
By asking the question of "do i need to slow down as the bit gets larger" it seems that you have an understanding of whats going on, and yes, for the bigger drill you do need to slow the RPMs compared to the pilot drill.
As for the chips, all I can say is that big drills suck. Its not that often that you'll see two nice shiny curls come off of one. In my case part of it is probably that big drills are expensive so I buy buy the cheapest piece of crap I can find.