How far do we have to go to leave the galaxy? How far is the sun from the edge of the galaxy in the following directions?


Moving along the shortest path parallel to the galactic plane (outward radially)
Moving along the longest path parallel to the galactic plane (radially presumably passing through the center)
Moving along the shortest path perpendicular to the galactic plane ("up")
Moving along the longest path perpendicular to the galactic plane ("down")


I realize these distances may be somewhat fuzzy (edge of galaxy is probably hard to define exactly, galaxy isn't exactly an ellipse, computing distances like this are hard even if the boundaries weren't fuzzy), so ranges and uncertainties are fine. I don't expect precision like computing whether certain points on the edge might be closer than just going straight out from the center and the like.
The sun is within a few parsec (15-25 pc) of the galactic plane, slightly above. The thin disk of the milky way (containing ~85% of the stars and gas) has a density going roughly like $\rho_0 \exp(-|z|/300 pc)$, while the thick disk (older stars, a few percent) has a scale height of 1000 pc instead. So if we want to move up (galactic north) above 90% of the star density we would have to move ~680 parsec, and going down 720 parsec. At this point at least the view would be splendid, but there would still be a fair number of thick disk and halo stars above and below. 

The disk has a scale length of 2.5–4.5 kiloparsec (kpc) and the sun is about 8 kpc away from the core, so we are already outside nearly 90% of the stars (not even counting the bulge). The radius is commonly given as 15.5-27.5 kpc, so we are 7.5-19.5 kpc from the nearest edge and 15.5-27.5 kpc from the far edge. 

If we count the dark matter halo the distances get even fuzzier. The virial radius (a rough measure of where most mass is inside most of the time) is 200 kpc, so we are basically 200 kpc away from that distance in most directions.