Your question, Jenny, requires an answer of greater magnitude and depth than we could possibly handle in this venue.
It has to do with differing approaches to economic theory dealing with the causes of poverty and its solution, which though they may have a general application, are influenced greatly by each local context, indeed by each individual.
For example, some people are poor through no fault of their own, whereas others are poor because they have not used their opportunities effectively.
As to "the teachings of Christianity," you will find almost as many answers as there are Christian economists and/or theologians. Here are some of my own thoughts.
1) The Bible says that we are to be sensitive to the needs of the poor. Both in the Old Testament as well as in the New, the people of God are encouraged to be generous with those who are suffering need. Jesus taught that we were to give to the one who asks (Mt 5:42).
2) Though we are to do good "to all people" as we have opportunity, Paul stresses that we are to put special focus on those whom we know who are of "the household of faith" (Gal 6:10). It is impressive that in the first days of the founding of the Church (Acts 4:34) the writer could note that "There were no needy persons among them" due to the generosity of their Christian friends. Paul also warns that the person who refused to work shouldn't eat! (2 Thes. 3:10)
3) Christians have also sought to go deeper and deal with endemic causes of poverty arising from injustice. For example, a by product of the Evangelical Revival in England in the 19th Century was the number of Christian leaders involved in creating labor unions, seeking to abolish the slave trade, push through prison reform, etc.
4) At the same time, poverty is not a simple thing, as I implied in my first paragraph. Much of the cause of poverty is just plain sin - in the lives of both the wealthy and the poor. An integrated understanding of the Gospel always needs to be understood, presented, and experienced.
5) The Bible never says that a rich individual cannot be truly "religious" (your term) or even a good Christian. Abraham was made rich by God. David was rich. Philemon undoubtedly was rich.
It is not being rich that is sinful; it is the insensitivity that wealth often brings, the sense of materialism and trusting in one's possessions that can so easily distract one from true spiritual wealth and dependence on God.
Jesus told the rich young ruler to dispose of his wealth because he trusted in it. Paul did not say this to Philemon, who opened his home and had a church functioning in it! It's not what we own, but who owns us!
Down through history the Lord has used wealthy and politically powerful Christians to accomplish things that they could not have done if they had been poor.
I know wealthy Christians who are exceedingly generous, and it would seem that the Lord continues to bless them with resources because they knew how to use them so effectively.
But...you and I need interchange, not a lecture!
I have given you some of my own ideas. As you continue reading the Bible note how the Lord gives us in His Word a balanced teaching on this deep and profound subject.
May He guide you to have "mercy on the poor" and use your possessions (as a "rich American") well.