Down means ‘from a higher part to a lower part of something’.
- The rain came down heavily.
- We ran down the road.
Down can also mean along.
- She walked down the road. (= She walked along the road.)
Note that to pay $10 down is to pay $10 now and the rest later.
Up means to or in a high(er) or (more) important place, degree etc.
- Pull your socks up.
- Are you going up to London (i.e. from the country) soon?
Trains to important places are often called up trains and trains from important places down trains. But note that in the US, downtown normally means in/to the central business/entertainment area.
People often use up for movements towards the north, and down for movements towards the south.