How does velocity affect the pace of time?

Your perception of time doesn't change if you travel even close to the speed of light. What changes is the perception that will have others respect of you. If they would be able to look at your watch they will see that it moves slower than theirs, and gets slower the more close to the speed of light you move. If you see their watches you will see that they are moving slower, because the movement is relative. It becomes absolute when a force is applied to it, for example an acceleration (the twins paradox). Time and space is relative to the individual that is moving at a certain speed. For you, a minute will always be a minute, you will perceive it as the same amount of time always.


---Addition---

The equations you are thinking of are the Lorentz Transformation, heres a link
http://www.classicalreality.net/Lorentz%20Transformation.htm

When you travel nearly the same speed of light, light keeps on traveling at the speed of light from your point of view. So it is not "hardly moving" but in fact keeps on moving for everybody, independent of their velocities, at 300,000 km/sec (in vacumm). The only explanation to this is that time for the different observers is not comparable, everybody follows its own time, and space also.

Moving any given object possessing mass requires energy. Speeding something up to near light speed requires enormous amounts of energy. According to Einstein, mass and energy are basically the same (e=mc²). So the more energy you put into a system, the more mass it acquires. This invariably results in more gravitation, and near the speed of light the gravitation is so massive that it literally bends the space-time continuum (examples for enormous gravitation are black holes).
The equations show that someone travelling at or near the speed of light would experience time (which isn't absolute but relative, thus part of our universe, thus subject to circumstance) going significantly slower than someone moving at the speed of, say, our galaxy (in other words, us).
This has been experimentally confirmed numerous times, although, of course,  we're talking tiny fractions of seconds here. But the equations set up by Einstein, so far as we know, correspond with physical reality.

It doesn't. Time is a manmade concept invented to deal with his/her concept of reality. As you perceive reality in this present moment your perception will give you one acceptable form of reality out of an infinite number of possible realities open to. Usually this perception is formed through memory and past conditioning. So your present is according to you dependent on your past; but also it can be dependent on your future if you are able to train your mind to accept that all your past and future experiences are all bundled into an infinite number of possibilities. Quantum theory explains this very well.