I would like to specify a few things before starting with my actual question :
My actual question is that, when we fly in a plane, for example, we won't be moving at $1000 \ \mathrm{mph}$ speed anymore right? Assuming your plane is going at $500\ \mathrm{mph}$, would time go 2 times slower?
No, time will not go 2 times slower.
$1000\ \mathrm{mph}$ speed is too slow compared to speed of light. Relativistic effects magnitude are about $(v/c)^2$, it's about $10^{-10}\ \%$ in this case.
Person $A$ stands on ground, person $B$ moves along with some speed $\implies$ person $A$ will see that person's $B$ clock tick slower. Person $B$ will see that his own clock ticks as usual, but he will see that person's $A$ clock tick slower. Sigh... They are both right.
I have to note, (and make things a little bit more confusing) that in case of time dilation on plane it is also necessary to account time dilation due to Earth gravity. Because of it time on plane goes a little bit faster (because it is high above the Earth).
For one thing, flying on a plane at 500 mph ground speed means you are going 500 mph faster than the ground is moving. So a plane flying East along the equator at that speed is moving at 1500 mph with respect to an observer on the pole. And as another user pointed out, the time dilation factor is not linear. Far from it actually. For smallish magnitudes it is approximated by $1-1/2v^2/c^2$, which for any terrestrial speeds normally attainable is indistinguishable from $1$ without very precise lab measurement equipment. That said, the effect is real, and even at very small speeds has been measured with precise enough equipment.
