Can someone confirm that this is how time dilation works?

[chestycougth]

I'm not sure if the way I understand it is correct or if it is slightly inaccurate, so I will just write it as I understand it and hope you can all my point out my errors.

As I understand it, there are 4 dimensions. 3 of space and 1 of time (first question. Is time actually considered an extra dimension?) which are collectively known as spacetime.
Your speed in spacetime is always equal to the speed of light (I will use 300,000,000 m/s for the sake of simplicity.)
If you are not moving at all in the first 3 dimensions then you will be moving at 300,000,000 m/s in time (second question. are distance units the right unit of measurement in this question?). If you are going at 150,000,000 m/s then time will go twice as slow as it would be as normal.

Also, what is the name of this quantity that space and time are shared between?

 

[PeterDonis]

As I understand it, there are 4 dimensions. 3 of space and 1 of time (first question. Is time actually considered an extra dimension?) which are collectively known as spacetime.

Yes, this is correct. Time is one of the 4 dimensions.

Your speed in spacetime is always equal to the speed of light (I will use 300,000,000 m/s for the sake of simplicity.)

This is one interpretation, but it can lead to problems if you try to take it too far.

If you are not moving at all in the first 3 dimensions then you will be moving at 300,000,000 m/s in time (second question. are distance units the right unit of measurement in this question?).

For the value you have given for the speed of light, you would measure distance in meters, like normal, and time in seconds, like normal. Relativity physicists often like to use units in which the speed of light is 1, such as time in years and distance in light-years, or time in seconds, and distance in light-seconds, or distance in meters and time in units of 3.3 nanoseconds (the time it takes light to travel 1 meter).

If you are going at 150,000,000 m/s then time will go twice as slow as it would be as normal.

Speed is relative; it depends on the observer. To yourself, you are always at rest, and you always perceive your time as flowing normally. Someone else, who saw you moving past them at half the speed of light, would see your time flowing more slowly.

Btw, the time dilation factor is given by the reciprocal of $\gamma = 1 / \sqrt{1 - v^2 / c^2}$. For $v = 1/2 c$, this gives $\gamma = 2 / \sqrt{3}$, so an observer who saw you flying past them at half the speed of light would see your time appear to flow at 0.866 of its normal rate (the rate it appears to you to flow).

 

[harrylin]

I'm not sure if the way I understand it is correct or if it is slightly inaccurate, so I will just write it as I understand it and hope you can all my point out my errors.

As I understand it, there are 4 dimensions. 3 of space and 1 of time (first question. Is time actually considered an extra dimension?) which are collectively known as spacetime.

Time can certainly be considered an extra dimension for calculations. See for a discussion:
http://bartleby.com/173/17.html
http://bartleby.com/173/a2.html

Because of that, there are many people who consider it to be an "actual" extra dimension.
Most of the rest has, I think, already been appropriately answered.

[..] Also, what is the name of this quantity that space and time are shared between?

Not clear what you mean; space-time interval perhaps?