Those are very good questions and they're quite hard to answer partially because they are put together from a number of pieces of information and indeed the actual answers as to exact shape and size are not agreed on
Firstly what we first called the milky way are stars in our galaxy but very far away in other arms. We know how far away they are because of what are called standard candles. These are stars who's beightness can be determined from their spectra. If you know how bright a star should be from it's spectra and you know how bright it appears to be to us you can work out it's distance.
You can also use the spectra to work out how fast it's moving towards or away from us
Mapping stars by their distance and speed you can start to work out what sort of shape the Galaxy must be.
I hate to refer you to Wikipedia but the article really is quite good
http://en.wikipedia.org/wiki/Milky_Way
As for other galaxies orbiting us - not quite sure that's the best way of explaining the situation.
There are a number of Galaxies in what is called the local Group. The Magallenic clouds and the quite beautiful Andromeda Galaxy for example. All of these are moving through space influenced by each others' gravities - so they're not really orbiting us so much.
Then there are Globular clusters which are clumps of stars orbitting galaxies. These are confusingly called population II stars confusing because they are first generation stars they're normally very old - mostly made of the Hydrogen and Helium from the big bang - whereas most in the galactic disk have many metals in them which was formed in earlier stars.
Way Way to much to say for an answerbank question
Of course it's difficult because we cant see a lot of it