It's a good idea to have it set to the highest resolution available without distorting the aspect ratio. But, more importantly, be sure that you have the hgihest amount of colors supported... that will usually read as 'True Color' in your display settings.
Between two different computers, the images will always look different... and they will never be all that close to how they will look when you print the photograph. This is a huge issue even for amateurs and professionals alike. The fact of the matter is that solving the difficulty of "how will this look when I print" or "how will this look on different screens" is incredibly difficult. Every monitor has a different idea of 'pure white' than the next. Every monitor has a slightly different idea of the optimal color settings. Thus, every monitor inevitably shows the image a little different than every other.
There are color meters designed specifically for calibrating your monitor to ensure the colors are as accurate as possible, but cheaper displays don't really even allow enough customizability to tweak the colors properly anyway. Most monitors that are even half-way affordable still can't properly display the full tonal range of RAW photographs, either.
As far as high-fidelity when printing... well, it's just as complicated. There's a specific technique, called 'soft-proofing', that involves getting profiles for your printer and the paper you will be using. Certain programs can then incorporate the qualities and color of the paper, as well as the properties of the ink, to give you a fairly accurate rendition of how your photo will look when it's printed. Even soft-proofing isn't always that accurate, though.
Without upgrading your monitor, buying a color meter, and acquiring printer and paper profiles... you're really just stuck making do with what you've got. Make sure your monitor is set to a nice, high-pixel count and that you are using True Color. It's the best you can really do.
People that make decent incomes off of photography still occassionally struggle with printing and display problems... it's just not that easy to contend with.