Would running along Earth's Surface at escape velocity cause you to get out of the Earth's gravitational pull? If one were to run/drive/fly along the earth's surface (by this I mean tangential to the Earth) at escape velocity (about 7 miles per second), would that person escape the gravitational pull? If not, at what speed would this happen? 
 A: Let's idealize things a little.  We'll pretend that earth's atmosphere is a vaccuum so as to forgo any issues with burning up from frictional forces.   We'll pretend that your kinetic motion is perpetual, so that the linearized velocity remains constant. As I see it currently, this means that the velocity you have would have to be so great as so your centripital acceleration would have to 'out do'  the downward acceleration due to the "force" due to earth's gravity.  (Let us bypass the discussion of whether or not gravity is a force for now.)  
Some approximate numbers/ formulas:
Radius of Earth is about 6400km
Mass of earth approximately 6*10^24 kg
Gravitational constant G is about 6.7*10^-11 m^3/kg/s^2
Escape velocity formula is $v_e=\sqrt{2GM\over r}$
Plugging in those numbers to the formula for earth this is about 11.2*10^3 m/s
So assuming you are  initially proceeding in this path in a circular motion that gives you a radial acceleration of
$a=v^2/r$
So if we plug in our number this gives about 19.6m/s^2 or close to twice the acceleration due to gravity at the earths surface (which is interesting).  So initially you would indeed tend to rise up off of the service of the earth if this speed were maintained.
Now the gravitational force falls of proportionally to $1\over{ r^2}$  But as your radius of distance from earths surface increases the centripetal acceleration only falls of like ${1\over r}$.  So, yes, under these idealized assumptions you would indee continue to get further and further away from the surface of earth assuming not running into a mountain or airplane.  Then again we are assuming a vacuum atmosphere so airplanes really shouldn't be an issue.
A: How can you run along the surface and escape, both at the same time?
You can not run faster that the orbital speed of a level, at any level. Escape velocity is higher than the speed of any orbit of earth.
However, if you start off with escape velocity, even in tangential direction, you will immediately lift off the surface. Then, atmosphere will slow/burn you down enough to fall back. You may even bump in some mountain ranges depending upon your starting location.
Ignoring the resistances, yes, you would escape earth. But then you are not running along the surface. 
A: Ignoring any intervening obstacles (such as mountains and skyscrapers) and assuming that your tangential velocity is high enough (from the surface of Earth) so that you wont crash into those obstacles, yes you would be able to escape Earth at some point but it wont be linear escape. This means that you will follow a sprial/helix trajectory until you are completely escaped. However, this escape will not be as efficient as vertical launches. 
A: A low Earth orbit takes about 90 minutes, so (ignoring mountains, air and earth's rotation) if you had a speed of about (40,000 km)/(7,200 sec) you could be in orbit just above the surface. If you started out with escape velocity (twice as fast) horizontally you would be on a parabolic orbit that would not return to Earth.
