Every time I take off my pullover the t-shirt below it comes up too and that's not really good looking. Is there a way to avoid this?
One way to take off your pullover is to start lifting it at the hip, and then kind of rolling it over your head, and finally pull out your hands. This method does however require you turn the pullover inside out afterwards.
As you lift it from the waist, instead of pulling it directly off from the top, any garment under usually stays put. You could tuck the t-shirt in your pants during the operation for extra protection, but most often it isn't necessary.
Take your arms out of the sleeves first. Then you can use your hands to keep your shirt on better. You can use one hand to keep it on and use one hand to take it off. Or you can push the shirt down or tuck it in and pull the pullover first. For these methods, if you are wearing a long sleeved shirt you can grab your sleeves so they don't come off.
Have a size shirt that fits you. To tight can be the difference in your shirt coming off.
Loosen the shirt from your other shirt first before attempting to take it off. Sometimes the shirts can get stuck together and they come off together.
Get someone to help you, if you think that it is going to be a really hard job. This may look weird, but you can always do it in a bathroom or away from the public.
First, I lift up the sweater just below my chest area so my shirt is exposed. Then, I move over towards a table/ledge of some sort, pin the shirt between my body and the ledge, and pull off the rest of the sweater. You can also, if for some reason there are no desks, tables, ledges, etc., lean against a wall and pin your shirt to it (curve your back or lean, if necessary).
Describing it makes it seem a lot harder than it is, but usually you can find a desk or something to do it on.
This is how I always take my sweater off without pulling my shirt up:
Pull arms from sleeves so they are inside. Then place fingertips below chin with palms facing in. Then lift sweater over head, while forearms and elbows push out from around waist, so shirt doesn't come up. Sweater ends up in front of you with your arms inside.