Also, does the exposure time change if the camera is shaken and it has to focused again? I did have an ISO of 3200 and shutter speed of 1/125. All I obtained was a dark sky. Its the first time I went looking for dark skies to capture starry skies.
Short answer: Tripod yes, remote no, but it is helpful.
Long answer: You will need a stable platform. A good tripod will work best. A remote is helpful, but not needed. You can also use the self timer. And you can configure some cams to switch the mirror into upward position some time ahead of the shutter to prevent shutter shock. All helping to take jitter-free shots.
As a starting setting, you will NEVER be at 1/125. Usually people start at 15-20 seconds. Be aware that at these times, stars will start to form star trails dues to the motion of earth.
To get ISO down, start at you widest aperture (the smallest number), 20 seconds, and ISO at around 6400. Then have a look what you will see. If too dark, go up with the ISO. If too bright, go down with ISO. If stars begin to trail, lower shutter speed to 15 sec, but you will then have to up the ISO.
Regarding focus: Set to manual focus and focus to the brightest star that you can see and let the focus there. Live View might be helpful there. If you point the cam elsewhere, you do not have to refocus.
There are whole websites giving tips for Astrophotography. This is the bare minimum. Please take some time and search for them to see what adventures await you there. For example there are star trackers that will move your cam according to the stars, so you can use longer exposures than 15-20 seconds. Have fun!