angie,
welcome.
Since you will starting off in London have you check what london has to offer?
From there you can look at what Paris and Amsterdam has that may interest you.
I have been to all three places and from what i like to do/see, i easlily spent 4 full days in both London and Paris and didnt see everything thats there. Those 4 full days were just in the city too, no day trips outside of the city. Amsterdam i spent 3 full days there and again no outside day trips.
YOu can do all 3 and more if you choose, but you should be aware that the more you choose the less you may see.
Figure out what and where you want to go most and go from there. If you can do alot of things in london while you are working there, then that may allow you more time outside of London.
as far as how to get around, you can do train or plane. If you decide to go to Paris from London, you have time to buy your tickets. The earlier the better since prices will be less. However, if your company will pickup the tab, then who cares?
happy trails.