The ISS has thrusters to avoid debris and to combat orbital decay. Do the astronauts feel the station accelerating? Do they have to hold onto something and is it the same when they rotate the station using reaction wheels for incoming spacecraft?
If you watch these videos:
...you can see that the acceleration is quite gentle, but definite. The astronauts do need to hang on to something if they don't want to drift to the back of whatever room they're in.
The first video was a reboost performed with the ATV service ship, as described in this article. Depending on what numbers you look at, acceleration during an ATV boost is something like 0.0035-0.005 m/s2 -- about 1/3000 or 1/2000 of a g. So an ounce or two of force (~30 grams-force) is enough to keep an astronaut in place -- easily done with a pinky finger. You can see in the first video that the astronauts can throw themselves to the opposite end of a compartment against the acceleration with little effort.
The second video is a reboost performed with the Zvezda module's two built-in thrusters, which are more powerful than ATV's. You can see that the acceleration is a little more rapid in that video; they say it's 0.0185 m/s2 -- still only 1/600 g.
ATV's thrusters yield about ~2kN of thrust. Progress's thrusters, also used for reboost, yield ~3kN; the Zvezda module's built-in thrusters about 6kN.
It's unclear to me how fast the station is turned by its control moment gyros, but I think it's usually less than 0.1 degree per second. They've done 90-degree and 180-degree rotations as tests, and the 180 (not done all at once) took 2 hours 45 minutes to complete. I think that would be harder to notice for a person near the center of the station.
The answer to the question, "Do the astronauts feel the station moving?" is yes, definitely, but sometimes in an "indirect" fashion.
During Space Shuttle mission STS-109, when floating in my sleeping bag and waiting for slumber to come, I would notice that occasionally my body would softly brush up against one side or the other of said sleeping bag. A quick glance at a laptop computer nearby, with, among other things, thruster status on its screen, would validate that a vernier jet (a small, 24 lb thrust attitude control jet) had just fired, in response to the autopilot, which was keeping the Orbiter in its previously-commanded attitude.
So, really, it was not a case of my body brushing up against the side of my sleeping bag, but a case of said sleeping bag (which was attached to the Orbiter) brushing up against my stationary body.
A measly little 24 lb thruster pushing around a 220,000 lb. spaceship.
The rate of change is pretty low with pretty low acceleration. (Consider the mass of the ISS(370,000 kilos) and the output of the Progress booster engines at just under 3000N of thrust).
One interesting experiment to consider is something you can try on a train that has open gangways. I.e. The cars are connected and it is one long tube. Stand at either the front or back of the train. When the train starts up from a station, pay attention to the breeze.
At rest, the air inside is at rest. (or is moving at the same speed as the earth rotates). As the train starts, the air stays still at first but the train moves around it. Then it piles up on the front of the train and bounces back towards the tail of the train.
Opposite when you stop. Toronto on the Yonge University line has these trains, and it is a pretty cool feeling once you realize what is happening.
Same happens in the ISS, but acceleration is much slower in comparison. The Toronto TR-1 subways accelerate at 0.9 m/s^2. There is no way the ISS accelerates that much.
They can definitely feel it when, like in January 2009, the Russians mess up and send crazy command data, resulting in too high of an acceleration and scary oscillations.
Video of the ISS rock and roll dance here - the burn is in progress when the video starts, and cuts off about halfway through. What I wouldn't give to have an exterior camera view of the solar arrays.