What would the Sagittarius A black hole look like if we were in a spaceship just outside its event horizon? Let's pretend that our spaceship is shielded well enough that being in the presence of the black hole won't kill us with radiation. I have read that the black hole is as large as the space between the Sun and the Earth. How large is its event horizon? Would the area around Sagittarius A be crowded with stars, or is the distance between them in the galactic core vast enough that its vicinity would be as serene as the space between celestial objects in our star system? Could you describe the gravitational forces in the vicinity just outside the event horizon?
 A: The black hole is 4.3 million solar masses. The event horizon (Schwarzschild radius) is $3(M/M_{\odot})$ km, so about 13 million km.
The answer to your next point can be found at Density of stars near the center of the Milky Way
The density of stars near the Galactic centre is thousands, heading up to hundreds of thousands per cubic parsec, compared with of order 0.1 per cubic parsec near the Sun. You just take the inverse cube root of this number to work out how close the nearest star will be on average. For the Sun this is $\sim$2 pc, at the Galactic centre it is $\leq 0.02$ pc. We can see about 5000 naked eye stars from Earth. Near the Galactic centre this could be increased by 4-5 orders of magnitude.
The gravitational field will be roughly given by $GM/r^2$ - though this is a Newtonian approximation. At say 10 times the Schwarzschild radius this would work out as about $3\times 10^6$ m/s$^2$. Closer to the black hole then your question would need refinement - you need to think about General Relativity and either the Schwarzschild or Kerr metrics.
Strong gravity in of itself is not readily apparent to an orbiting body and tidal forces are relatively modest around such a massive body. However, over the course of time these weak tidal forces may be capable of eventually drawing stars inwards should they approach this close to the black hole, though gas is much more effectively accreted.
EDIT: Here is a picture of the Galactic centre taken with an infrared telescope. Sagittarius A* is in the middle. Some of the stars in this image might be in the foreground, but many have been observed in their orbital motion around the black hole (indicated by ellipses). Many, many stars are unobservable because they are too faint, only the most massive will be seen here. At the distance of Sagittarius A*, this picture is about 0.08 parsecs across (the scale on the picture is in arcseconds) and appears to contain at least 100 massive stars. Even if we only include the dozen or so objects known to orbit within about 0.05 pc of the centre, this leads to a density of 100,000 stars per cubic parsec.

