Learn Russian: Мы используем новую программу.

The verb использовать (to use) belongs to a special group of verbs ending in -овать. When you conjugate them in the present tense, the -ова- part changes to -у- before adding the endings. So, for мы (we), it becomes используем instead of использоваем.

Why are the words for new program spelled новую программу?

They are in the accusative case because they are the direct object receiving the action of the verb. For feminine words, the adjective ending -ая changes to -ую (новую), and the noun ending -а changes to -у (программу).

This lesson is about который (which). How would I add the clause "which is interesting" to the end of this sentence?

You would say: Мы используем новую программу, которая интересная. Even though новую программу is in the accusative case in the main sentence, the word которая is the subject of its own relative clause. Therefore, которая takes the nominative case, matching the feminine gender of программа.

Does используем mean we use or we are using?

It can mean both! Russian doesn't have a separate continuous tense like English does. Depending on the context, мы используем translates perfectly to either we use (regularly) or we are using (at this very moment).