в чем разница?

[LizzyBennet]

Как правильно: we are going down Tverskaya Street или we are going up Tverskaya Street? И в чем вообще разница между to go down the street и to go up the street?

[asil]

Как правильно: we are going down Tverskaya Street или we are going up Tverskaya Street?

The difference depends on several things
1. Whether the street has a slope.
2. Which end you personally perceive as 'up' - for different reasons, e.g. the end of the street you are facing would be 'up'.
However, as a rule, you go 'up the street' towards the place which is busy and lively, e.g. if you are near Pushkinskaya (or whatever it is called now), going up the street would be in the direction of the Red Square.
Also, we do know that the Kremlin, as any fort, is supposed to be on a hill. So, going up Tverskaya St is quite justifiable here.
3. If you are moving in the direction where the street numbers (addresses) increase, then again you are walking 'up the street'.

But I still think that 'going down the street' is by far more common. If in doubt, you can always (almost!) say 'go along the street'. 

[nas2000]

No difference.
Up/down/along Tverskaya Str.

[chaika]

In Russian you use both?
Эх, вдоль по Питерской, по Тверской-Ямской, Да, ох, ой, по Тверской-Ямской да с колокольчиком.