There are some wildly speculative but not technically impossible design studies for interstellar probes. One is Project Daedalus, which would use pulsed fusion to reach 0.1 C. That’d only take ~5,000 years to leave the galaxy, and another 500,000 to get a wide-angle shot. Of course, such designs are hardly practical, and they tend to assume things like a large scale Jupiter helium-3 mining operation for the fuel.
You’d still need a snack.
From here (1995)
http://adsabs.harvard.edu/full/1995AJ....110.2183H
the Sun is reckoned to be 20.5 +/- 3.5 parsecs above the galactic midplane. That is, we are somewhere between 55 and 78 light years above the midline; in a galaxy that extends 500 light years above and below the plane, that is not really going to make much difference in the observed density of stars in either direction.
The escape speed from our galaxy for an object at Earth’s distance is somewhere between 450 and 600 km/s. (This is not trivial to calcluate because unlike Earth or Solar escape speed the mass of the galaxy cannot be treated as concentrated at a point.) By comparison, escape speed from the Sun at Earth’s orbit is about 42 km/s, which was achieved by the Voyager probes only by additional momemtum acquired by swing-by maneuvers at Jupiter and Saturn. Even neglecting losses due to drag of the interstellar medium, it would require some technomagical propulsion solution to even achieve an elliptical orbit sufficiently above the galactic plane to image the entire galaxy. And of course, the distance and intervening gas would be such that transmitting a signal back to Earth (provided you could still track its location at that time and distance) would be impossible without a communication source with the output power of a modest supernova.
We can map our galaxy “indirectly” i.e. by spectrophotometry and creation of a three dimensional map as is being done with the ESA Gaia mission, but we’ll only be able to directly observe a small portion of our galaxy; Gaia is mapping less than 1% of the luminous objects in the Milky Way due to occlutation by dust. (The dark band you see sandwiched by the two relatively bright streams that make up the Milky Way is actually dust in the galactic plane which prevents us from observing most objects.) The vast majority of our own galaxy is less visible to us than objects millions of light years away.
Stranger