How to learn to score new documents based on a existing set of scored documents?
First come up with some features for the document. Stuff like frequency of some popular words associated with that topic might work. In this way get the features for all the documents and then apply some algorithm. Some ways you could apply them are:

1) k means - cluster the documents on the basis of the features. Each cluster should be predominantly associated with a particular score value. Then see which cluster a new document will be assigned to.

2) Supervised learning - use neural networks, multiclass SVMs etc to classify the new document to a particular class (score) using the model you would have generated.

All of these are examples of classification to a discrete score value. However, since you are dealing with a large score range (0-2000), you could also try something like regression which will give you a continuous value, but could be rounded off to the nearest discrete one.

Check out the Coursera course on Machine Learning for a great introduction!