How do Antonio and Shylock's religious beliefs and business approaches affect their relationship in act 1?

Hmm. The relationship between Antonio and Shylock stretches beyond Act I, but if we limit ourselves to that act, we see the following:
Antonio needs Shylock, and, to a lesser degree, Shylock needs Antonio. There is therefore some degree of mutuality. They are also similar in that Antonio makes a point of saying he has not put all of his investments aboard one boat, but has spread the risk. Likewise, Shylock loans to many people. In these areas they are similar.

However, they are also similar in their dislike for one another, which is based in part on their religions, or at least, on the prejudices their time educates into them regarding their religions. Both also put on something of a false front when dealing with one another.

We know that Shylock drops this false front when alone, and that Antonio drops it when with Bassiano and Gratiano. When Antonio is with Bassiano, we see different motives for investment. Shylock lends money almost maliciously; he seeks to have people (at least Antonio) at his mercy. Bassiano asks for money due to friendship, and Antonio stands for it. They have, therefore, different motives in all of this.

How do Antonio and Shylock's religious beliefs and business approaches affect their relationship in act 1?

Shylock and Antonio resent and dislike one another. Firstly, Shylock hates Antonio because he is a Christian; secondly, because Antonio is a Christian and therefore not allowed to charge interest, Antonio undercuts Shylock’s business by lending money without charging interest. The fact that Shylock does loan money and charges interest is another reason for Antonio not to like him. As a businessman, Antonio must resent Shylock's ability to earn extra money by charging interest.

Because Shylock is a Jew and lends money charging interest, Antonio feels morally superior towards him. He uses any opportunity to show his scorn and publicly humiliate Shylock. This leads to Shylock resenting and hating Antonio even more.

Antonio for example has publicly spit on Shylock. When Shylock sees an opportunity to take revenge on Antonio he takes it. The both engage in trying to annoy the other. For example, to close the deal with Shylock, Antonio invites him dinner. Shylock refuses with the reason that he is Jewish and the food would not be kosher (Jews don't eat pork) however, it is seen as a social slight to not even come and be present.

I hope this helps... look at the links below for a more detailed analysis.

How do Antonio and Shylock feel about each other's religions in act 1, scene 3 of The Merchant of Venice?

Antonio, in contrast to his warm and generous feelings toward Bassanio, feels hatred and contempt for Shylock's being a Jew. We learn from his conversation with Shylock that Antonio has a history of speaking in a cruel and derogatory way toward Shylock. He's even spit on Shylock. As Shylock notes:

Signor Antonio, many a time and oft

In the Rialto you have rated me

About my moneys and my usances.

Still have I borne it with a patient shrug,

For sufferance is the badge of all our tribe.

You call me misbeliever, cutthroat dog,

And spet upon my Jewish gaberdine.

I am as like to call thee so again,

To spet on thee again, to spurn thee too.

If thou wilt lend this money, lend it not

As to thy friends, for when did friendship take

A breed for barren metal of his friend?

But lend it rather to thine enemy,

Who, if he break, thou mayst with better face

Exact the penalty.

How like a fawning publican he looks!

I hate him for he is a Christian,

But more for that in low simplicity

He lends out money gratis and brings down

The rate of usance here with us in Venice.

If I can catch him once upon the hip,

I will feed fat the ancient grudge I bear him.

How do Antonio and Shylock feel about each other's religions in act 1, scene 3 of The Merchant of Venice?

It is clear that there is little love lost on both sides of this relationship. Clearly Shakespeare intends Shylock to be a representative of the Jews, and likewise, to Shylock, Antonio becomes a representative of Christianity and the way that it has consistently mistreated Jews down through the ages. Note what Antonio says to Bassanio after Shylock tells the story about Jacob:

The devil can cite Scripture for his purpose.

An evil soul producing holy witness

Is like a villain with a smiling cheek...

He goes on to say to Shylock that he should lend the money to "thine enemy" rather than to his friend.

Shylock is even more overt about his feelings towards Antonio. Note how, as he watches Antonio draw near, he says, in an aside:

I hate him for he is a Christian.

He plans immediately to "feel fat the ancient grudge I bear him" and somehow to get his own back for the way that he lends money without charging interest and also for the personal wrongs that Antonio has committed against him.

In this scene, therefore, Shakespeare introduces us to the main conflict of the play and we can see its roots and its development in the "merry bond" that Shylock proposes and Antonio accepts.