Why does this error occur?

From the Java documentation an IndexOutOfBoundsException is:

Thrown to indicate that an index of some sort (such as to an array, to a string, or to a vector) is out of range.

(the hint is in the name).

Your list has only two elements and so the only indices 0 to 1 can be referenced without throwing this exception.

How can I handle dynamic lists properly and avoid such errors?

Check the list size before trying to retrieve an element at a given index. For list

ArrayList<String> myList = new ArrayList<>();

the maximal index that can be accessed without an exception being thrown is

myList.size()-1

because indices start at 0 and there is a unique index for every element. Use an if-statement to check whether the index you are trying to access is within bounds:

if (index < myList.size()) {
    // Get value at in-bounds index
    fruit = myList.get(index)

    // Set value at in-bounds index
    myList.set(index, "Orange")
}

Is there a better way to check if an element exists before trying to access it?

In practice, it is rare to operate over specific indices in lists the way you've done in your code. As a data structure, lists work best when we want to iterate over a collection of elements and perform some operation over some or all of them (e.g. filtering and transformation).

// Filtering example
List<String> filteredList = new ArrayList<>();
for (String fruit: myList) {
    if fruit.endsWith("berry") {
        filteredList.add(fruit);
    }
}

// Mapping example
List<String> mappedList = new ArrayList<>();
for (String fruit: myList) {
    mappedList.add(fruit.toLowerCase());
}

If you wish to inspect and modify particular elements, a hash table may work better. This data structure is implemented as a HashMap in Java. It associates every value with a unique key. The get(Object key) method can be used for safe access to elements:

get(Object key) ... Returns the value to which the specified key is mapped, or null if this map contains no mapping for the key.

Below is a visual example of an 8-bucket hash table:

+------------------+
    |    Hash Table    |
    +------------------+
Index|   Key  | Value  |
-----|--------|--------|
  0  | null   |  null  |
-----|--------|--------|
  1  |"apple" |  "red" |---> ["grapefruit","orange"] ---> ["avocado","green"]
-----|--------|--------|
  2  | null   |  null  |
-----|--------|--------|
  3  |"banana"| "yellow"|
-----|--------|--------|
  4  | null   |  null  |
-----|--------|--------|
  5  |"mango" |"orange"|---> ["strawberry", "red"]
-----|--------|--------|
  6  | null   |  null  |
-----|--------|--------|
  7  |"grape" |"purple"|
-----|--------|--------|

Legend:
- Empty buckets shown as null
- Indices 1 and 5 shows collision of keys with chaining
- Each bucket stores fruit and its color
- ---> represents linked list for collision resolution