If we could capture an asteroid or meteor, can we use it as a vehicle for space travel? From what I understand, propulsion is a result of mass flow, so we could use a rail gun configuration to propel the rock mass from the asteroid or meteor. The asteroid could also provide some protection from flying into micro meteors. Is this a possibility or am I a dreamer? (I am an absolute novice.)
There have been many discussions about using asteroids for reaction mass. One obvious way would be to process material from icy asteroids into rocket fuel. Much of them are largely water, and that gives hydrogen and oxygen. If a carbon source were also accessed (like a carbonations contrite) then methane could be made. Or some may contain frozen methane. An obvious way to do this would be to transport small asteroids to L5 in the earth / moon system and process them there. Another way would be to use any solids as reaction mass, fired from a mass driver (magnetic rail gun). Even if large reusable rockets become common, the cost of lifting fuel will continue to encourage people to find sources that are already in orbit.
The main problem with using the asteroid as reaction mass is that you need a large power source. Science fiction stories usually assume that a nuclear fusion (or even anti-matter) reactor is available to heat up the asteroid material to a high enough temperature so that its momentum can propel the asteroid.
In reality it will be quite a long time before such power sources become available - if ever. Until then, the only possible power source is solar power. Near earth, approximately $1 kW$ is available per square metre. As the asteroid belt is more than twice as far away from the sun as the earth is, the available power there is less than a quarter, or less than $250W/m^2$.
Now let's do some calculations. Assume an ice-based asteroid of 50 m diameter. It would have a mass of about 500,000 tons. To accelerate this at $1m/s^2 $ we need a thrust of 500,000 tons. The Saturn first stage engines generated a total of about 750 tons thrust or 40 MW each.
Allowing for some inefficiency, let's assume we need 50 MW to generate 500 tons of thrust, or 40 GW for to achieve our $1m/s^2$ acceleration. We would then need a solar panel with a total area of $160,000,000 m^2$, a square of more than 12 km on a side. Reducing the acceleration to $1mm/s^2$ still leaves us with a square of 400 m.
Then we need to ship all this to the asteroid. At $100kg/m^2$ for the solar panel, we are talking about sending 1,600 tons to the asteroid.
All this may become feasible in the distant future, but don't hold your breath.