If you said only "je lis", the "s" is not pronounced. If you said "Je lis un livre", there would be a liaison ("Z" sound) added between the two words. http://french.about.com/library/pronunciation/bl-liaisons.htm
The "am" in "I am reading" is part of the English Present Continuous tense. French has only one present tense, whereas English has two (simple present and present continuous). But your identification of "am" as a separate verb helps explain why some learners want to focus on translating it separately into French--but it doesn't work that way.
- je lis = I read (or) I am reading
- tu lis = you read (or) you are reading
You can see a more detailed explanation and other examples in the Tips & Notes found at the beginning of this section.
French does not have a present continuous tense, only present tense. It may be translated to English either in the simple present (I read) or the present continuous (I am reading). To the French, it is just "je lis". However, when speaking in French and you want to emphasize that the action is going on right at this moment, you may use a different expressions (en train de). It works like this: Je suis en train de lire = I am reading (i.e., I'm right in the middle of doing it).