String Array object in Java

First, as for your Athlete class, you can remove your Getter and Setter methods since you have declared your instance variables with an access modifier of public. You can access the variables via <ClassName>.<variableName>.

However, if you really want to use that Getter and Setter, change the public modifier to private instead.

Second, for the constructor, you're trying to do a simple technique called shadowing. Shadowing is when you have a method having a parameter with the same name as the declared variable. This is an example of shadowing:
----------Shadowing sample----------
You have the following class:

public String name;

public Person(String name){
    this.name = name; // This is Shadowing
}

In your main method for example, you instantiate the Person class as follow:
Person person = new Person("theolc");

Variable name will be equal to "theolc".
----------End of shadowing----------

Let's go back to your question, if you just want to print the first element with your current code, you may remove the Getter and Setter. Remove your parameters on your constructor.

public class Athlete {

public String[] name = {"Art", "Dan", "Jen"};
public String[] country = {"Canada", "Germany", "USA"};

public Athlete() {

}

In your main method, you could do this.

public static void main(String[] args) {
       Athlete art = new Athlete();   

       System.out.println(art.name[0]);
       System.out.println(art.country[0]);
    }
}

Currently you can't access the arrays named name and country, because they are member variables of your Athelete class.

Based on what it looks like you're trying to do, this will not work.

These arrays belong in your main class.

Your attempt at an athlete class seems to be dealing with a group of athletes, which is a design fault.

Define a class to represent a single athlete, with fields that represent the athlete's attributes:

public class Athlete {
    private final String name;
    private final String country;
    private List<Performance> performances = new ArrayList<Performance>();
    // other fields as required

    public Athlete (String name, String country) {
        this.name = name;
        this.country = country;
    }
    // getters omitted

    public List<Performance> getPerformances() {
        return performances;
    }

    public Performance perform(Dive dive) {
        // not sure what your intention is here, but something like this:
        Performance p = new Performance(dive, this);
        // add new performance to list
        performances.add(p);
        return p;
    }
}

Then your main method would use ti like this:

public class Assignment1 {
    public static void main(String[] args) {
        String[] name = {"Art", "Dan", "Jen"};
        String[] country = {"Canada", "Germant", "USA"};
        Dive[] dive = new Dive[]{new Dive("somersault"), new Dive("foo"), new Dive("bar")};
        for (int i = 0; i < name.length; i++) {
            Athlete athlete = new Athlete(name[i], country[i]);
            Performance performance = athlete.perform(dive[i]);   
            // do something with athlete and/or performance
        }
    }
}