According to the Catholic Church, who taught Moses the Law? Who taught Moses the Law? (Meaning the Law given to him on Sinai, that he would teach to all of Israel.)

Judaism says it was angels. What does the Catholic Church say?
TL:DR: Regarding the old law, God gave the law to his people.

In the case of delivering the Law to Moses at Sinai, Exodus 19-24 is pretty clear on who delivered what.   

For a longer answer to your question, @Marc has suggested Covenant and Kinship by Scott Hahn.  I second that recommendation, because as Marc points out (in a comment) the answer to your question is related to Covenant theology.  

Who gave the law to Moses?

Beyond the scripture cited in the TL:DR, the Catechism of the Catholic Church articles 1949 - 2051 discuss divine law, moral law, the old law, and the new law.  Within the discussion of the old law it is pretty clear that the simple answer is that God gave the law to Moses.  That is the teaching of the church.   

I recommend reading those articles to take your understanding of the Church's stated belief beyond a sound byte.  It's a shorter read than Hahn's book, but it's not as in depth.  Each page and each article have footnotes indicating the scripture, theologian, or church father supporting a presented belief.  That notation is very helpful, as it provides fertile ground for further reading and understanding of the position.    

The law in general:


  1952 There are different expressions of the moral law, all of them
  interrelated: eternal law - the source, in God, of all law; natural
  law; revealed law, comprising the Old Law and the New Law, or Law of
  the Gospel; finally, civil and ecclesiastical laws.
  
  1953 The moral law finds its fullness and its unity in Christ. Jesus
  Christ is in person the way of perfection. He is the end of the law,
  for only he teaches and bestows the justice of God: "For Christ is the
  end of the law, that every one who has faith may be justified."4
  (Romans 10:4)  


The Old Law


  1961 God, our Creator and Redeemer, chose Israel for himself to be his
  people and revealed his Law to them, thus preparing for the coming of
  Christ. The Law of Moses expresses many truths naturally accessible to
  reason.  These are stated and authenticated within the covenant of
  salvation.
  
  1962 The Old Law is the first stage of revealed Law. Its moral
  prescriptions are summed up in the Ten Commandments. The precepts of
  the Decalogue lay the foundations for the vocation of man fashioned in
  the image of God; they prohibit what is contrary to the love of God
  and neighbor and prescribe what is essential to it. The Decalogue is a
  light offered to the conscience of every man to make God's call and
  ways known to him and to protect him against evil: 
  
  God wrote on the tables of the Law what men did not read in their
  hearts.1




Notes:
Since you did not provide a source for "what the Jews teach" in terms of angels delivering the law, I will not address that.  

1 St. Augustine, En. in Ps. 57, 1: PL 36, 673.