At What Point Does Mass Cease to Matter? I know very little about physics so I naturally have a lot of simple minded questions. 
Assuming I am piloting a rocket and want to obtain the fastest speed possible. 


*

*At what point does mass more or less cease to influence drag/retrograde effects on the rocket? 

*When I am "out of" (more or less free from) the gravity well of the Earth? 

*Or maybe it is minimal as long as I am outside of the atmosphere?
 A: *

*Mass affects the acceleration through $f=ma$, The higher the mass, the lower the acceleration for a given force. Drag is not affected by mass, only by the speed and the density of the atmosphere. Mass never ceases to influence the rocket's behaviour. Any change in speed or direction (i.e. any change in velocity) is affected by mass through the above formula.

*You are never out of the gravity well of earth, it stretches  infinitely far. However, if you go more than $11km/s$ directly away from earth, you will continue to move away from it to infinity. This assumes there are no other bodies present: $11km/s$ certainly will not let you escape the sun. In other words, you will stay in the solar system.

*Not sure what you mean by "it is minimal". Mass and gravitation are  not affected by the atmosphere at all - drag is, of course. Mass always stays the same, gravitational force (from the earth) reduces as you move further from the earth.

A: Simple questions are good questions, and often do not have simple answers.
Gravity will always effect your rocket, although if you're far away enough from the gravity's source it may be negligible (not noticeable to you). So how far is "enough"? It depends on your limit of precision. Suppose you were making a trip that took 10 days, and you wanted to get there +- 1 meter, then you can neglect velocities that are smaller than 1/10 m days$^{-1}$. You can then neglect accelerations that are smaller than 1/100 m days$^{-2}$, and thus forces that are smaller than $m_{rocket}$/100 kg m days$^{-2}$. You can then use this to find the distance from the gravity source at which the force is smaller than your limit, by rearranging Newtons law of gravity for r.
$$
F = \frac{GMm}{r^2}, r = +\sqrt\frac{GMm}{F} = + \sqrt\frac{GM}{a}
$$
E.g. For Earth (M = 5.972 × 10$^{24}$ kg) and a space shuttle with a mass of 2000 tons like discovery, r works out as 
$$
r = \sqrt \frac{ 6.674×10^{−11} * 5.972 × 10^{24}}{1/(100*24*60*60)} = 5.87 × 10^{10}m
$$
For a sense of scale, the distance from the earth to the sun is 1.496 × 10$^{11}$m.
N.B. In reality, there will be so many more sources of gravity and error that you'll want a much higher precision.
