try {
    StyledDocument doc = new DefaultStyledDocument();
    doc.insertString(0, "This is the beginning text", null);
    doc.insertString(doc.getLength(), "\nInserting new line at end of doc", null);
    MutableAttributeSet attrs = new SimpleAttributeSet();
    StyleConstants.setBold(attrs, true);
    doc.insertString(5, "This is bold text after 'this'", attrs);
} catch (BadLocationException ex) {
    //handle error
}

DefaultStyledDocuments will probably be your most used resources. They can be created directly, and subclass the StyledDocument abstract class.

try {
        JTextPane pane = new JTextPane();
        StyledDocument doc = new DefaultStyledDocument();
        doc.insertString(0, "Some text", null);
        pane.setDocument(doc); //Technically takes any subclass of Document
    } catch (BadLocationException ex) {
        //handle error
    }

The JTextPane can then be added to any Swing GUI form.

StyledDocuments generally do not implement clone, and so have to copy them in a different way if that is necessary.

try {
        //Initialization
        DefaultStyledDocument sourceDoc = new DefaultStyledDocument();
        DefaultStyledDocument destDoc = new DefaultStyledDocument();
        MutableAttributeSet bold = new SimpleAttributeSet();
        StyleConstants.setBold(bold, true);
        MutableAttributeSet italic = new SimpleAttributeSet();
        StyleConstants.setItalic(italic, true);
        sourceDoc.insertString(0, "Some bold text. ", bold);
        sourceDoc.insertString(sourceDoc.getLength(), "Some italic text", italic);
        
        //This does the actual copying
        String text = sourceDoc.getText(0, sourceDoc.getLength()); //This copies text, but loses formatting.
        for (int i = 0; i < text.length(); i++) {
            Element e = destDoc.getCharacterElement(i); //A Elment describes a particular part of a document, in this case a character
            AttributeSet attr = e.getAttributes(); //Gets the attributes for the character
            destDoc.insertString(destDoc.getLength(), text.substring(i, i+1), attr); //Gets the single character and sets its attributes from the element
        }
    } catch (BadLocationException ex) {
        //handle error
    }

Using the AdvancedRTFEditorKit library you can serialize a DefaultStyledDocument to an RTF string.

try {
   DefaultStyledDocument writeDoc = new DefaultStyledDocument();
   writeDoc.insertString(0, "Test string", null);
      
   AdvancedRTFEditorKit kit = new AdvancedRTFEditorKit();
   //Other writers, such as a FileWriter, may be used
   //OutputStreams are also an option
   Writer writer = new StringWriter();
   //You can write just a portion of the document by modifying the start
   //and end indexes
   kit.write(writer, writeDoc, 0, writeDoc.getLength());
   //This is the RTF String
   String rtfDoc = writer.toString();
     
   //As above this may be a different kind of reader or an InputStream
   StringReader reader = new StringReader(rtfDoc);
   //AdvancedRTFDocument extends DefaultStyledDocument and can generally
   //be used wherever DefaultStyledDocument can be.
   //However for reading, AdvancedRTFDocument must be used
   DefaultStyledDocument readDoc = new AdvancedRTFDocument();
   //You can insert at different values by changing the "0"
   kit.read(reader, readDoc, 0);
   //readDoc is now the same as writeDoc
} catch (BadLocationException | IOException ex) {
    //Handle exception
    ex.printStackTrace();
}