Time is a universal phenomena in the sense that we all experience time (just as we all experience the 3 dimensions of space) but it might not be the same time everywhere (just as you're not in the same place every where you are).

Time does go at different rates in different places/situations (slower while traveling near the speed of light or while being near a large gravitational pull) but the room around you is probably always going to be in the same situation (ex: if your spaceship is going 80% of c the whole spaceship will experience the same time dilation) so interacting with that room will feel the same as if it were in a different situation (ex: if you look at a clock on that spaceship it will appear to go at the same rate as if you were looking at a clock on Earth). The only time this breaks apart is while entering a black hole, where both time and space dilate, but they do so at a greater rate to your feet than your head (assuming you're falling in feet first). In that situation, your head might feel normal for a few moments while your legs would be being stretched at the molecular level and slowed down relative to your head. To someone on the outside, however, it would appear as if you had frozen on the event horizon because of the time difference.