The simplest way I can think of is simply to walk through the document.
Document document = editor.getDocument();
try {
String find = //... String to find
for (int index = 0; index + find.length() < document.getLength(); index++) {
String match = document.getText(index, find.length());
if (find.equals(match)) {
// You found me
}
}
} catch (BadLocationException ex) {
ex.printStackTrace();
}
You could write some routines to "find the next word" based around this concept
You may also find Highlight a word in JEditorPane and Java - Scroll to specific text inside JTextArea which both use a similar approach to achieve different things