To my understanding of history, England forcibly converted Ireland to Catholicism and then under the rule of Henry VIII England became protestant, so after that England discriminated against Ireland since they were Catholic and England was Protestant. Is this what really happened, or is there more or less to it?
The second part is basically right, Catholics were discriminated against after the reformation. Even after the reformation however there were English Catholics and Irish Protestants. In 1766 a process of Catholic Emancipation began, dismantling the laws which placed restrictions on them, the most significant of which was the Catholic Relief Act of 1829. There was however widespread cultural prejudice against Catholics and the Irish in particular which would last long after this.
The first part though, not at all. The Christianisation of Ireland happened before Anglo-Saxon England. Famously associated with the missionary work of St Patrick in the second half of the fifth century. Irish monks actually played an instrumental role converting England to Catholicism in the Hiberno-Scottish missions (the earliest of which started in 563).