In Gödel's Ph.D. thesis, published in 1930, he proved the Completeness Theorem for first-order logic. By the end of 1930, Gödel also proved his more famous Incompleteness Theorem. This is the original paper, translated and annotated by van Heijenoort. I have some copies of this. Let me know if you would like one. This is not required reading, just for your interest.
Also, here is an on-line translation and annotation and even changing to modern notation of the above paper. The translator is named Martin Hirzel. I have not gone through this in detail, but it looks very readable and nice! If you read it and have comments or find errors please let me know.