This should do the job:

public bool DoSearch(RichTextBox richTextBox, string searchText, bool searchNext)
{
  TextRange searchRange;

  // Get the range to search
  if(searchNext)
    searchRange = new TextRange(
        richTextBox.Selection.Start.GetPositionAtOffset(1),
        richTextBox.Document.ContentEnd);
  else
    searchRange = new TextRange(
        richTextBox.Document.ContentStart,
        richTextBox.Document.ContentEnd);

  // Do the search
  TextRange foundRange = FindTextInRange(searchRange, searchText);
  if(foundRange==null)
    return false;

  // Select the found range
  richTextBox.Selection.Select(foundRange.Start, foundRange.End);
  return true; 
}

public TextRange FindTextInRange(TextRange searchRange, string searchText)
{
  // Search the text with IndexOf
  int offset = searchRange.Text.IndexOf(searchText);
  if(offset<0)
    return null;  // Not found

  // Try to select the text as a contiguous range
  for(TextPointer start = searchRange.Start.GetPositionAtOffset(offset); start != searchRange.End; start = start.GetPositionAtOffset(1))
  {
    TextRange result = new TextRange(start, start.GetPositionAtOffset(searchText.Length);
    if(result.Text == searchText)
      return result;
  }
  return null;
}

The reason for the for() loop in FindTextInRangeUnfortunately the range.Text strips out non-text characters, so in some cases the offset computed by IndexOf will be slightly too low.

I have used a Different Approach. Using a TextBox to Set the Keyword; this Searches for the KeyWord on Button Click. Finds the Keyword; Highlighs it and Focuses on that KeyWord.

    // Index of Current Result Found (Counts Characters not Lines or Results)
    private int IndexOfSearchResultFound;
    // Start Position Index of RichTextBox (Initiated as 0 : Beggining of Text / 1st Char)
    private int StartOfSelectedKeyword;
    private int EndOfSelectedKeyword;

    private void btnSearch_Click(object sender, EventArgs e)
    {
        // Reset Keyword Selection Index. (0 is the Staring Point of the Keyword Selection)
        IndexOfSearchResultFound = 0;

        // Specify the End of the Selected Keyword; using txt_Search.Text.Lenght (Char Ammount).
        EndOfSelectedKeyword = txt_Search.Text.Length;

        // If txt_Search.Text is not Empty
        if (txt_Search.Text.Length > 0)
        {
            // Find Keyword in RichTextBox.Text
            IndexOfSearchResultFound = FindKeyword(txt_Search.Text.Trim(), StartOfSelectedKeyword, rtb_Hosts.Text.Length);

            // If string was found in RichTextBox; Highlight it and Focus on Keyword Found Location
            if (IndexOfSearchResultFound >= 0)
            {
                // Focus on Currently Found Result
                rtb_Hosts.Focus();

                // Highlight the search string
                rtb_Hosts.Select(IndexOfSearchResultFound, EndOfSelectedKeyword);

                // Sets a new Starting Position (after the Position of the Last Result Found)
                // To be Ready to Focus on the Next Result
                StartOfSelectedKeyword = IndexOfSearchResultFound + EndOfSelectedKeyword;
            }
        }
    }


    private int FindKeyword(string _SearchKeyword, int _KeywordSelectionStart, int _KeywordSelectionEnd)
    {
        // De-Select Previous Searched String (Keyword)
        if (_KeywordSelectionStart > 0 && _KeywordSelectionEnd > 0 && IndexOfSearchResultFound >= 0)
        { rtb_Hosts.Undo(); }

        // Set the return value to -1 by default.
        int retVal = -1;

        // A valid Starting index should be specified.
        // if indexOfSearchText = -1, Means that Search has reached the end of Document
        if (_KeywordSelectionStart >= 0 && IndexOfSearchResultFound >= 0)
        {
            // Find Keyword
            IndexOfSearchResultFound = rtb_Hosts.Find(_SearchKeyword, _KeywordSelectionStart, _KeywordSelectionEnd, RichTextBoxFinds.None);

            // Determine whether the text was found in richTextBox
            retVal = IndexOfSearchResultFound;
        }
        // Return the index to the specified Keyword text.
        return retVal;
    }

The only thing I couldn t achieve yet is to return to the 1st Search Result

This is a variant that will find the match closest to the Caret position.

  private TextRange FindText(string findText)
    {
      var fullText = DoGetAllText();
      if (string.IsNullOrEmpty(findText) || string.IsNullOrEmpty(fullText) || findText.Length > fullText.Length)
        return null;

      var textbox = GetTextbox();
      var leftPos = textbox.CaretPosition;
      var rightPos = textbox.CaretPosition;

      while (true)
      {
        var previous = leftPos.GetNextInsertionPosition(LogicalDirection.Backward);
        var next = rightPos.GetNextInsertionPosition(LogicalDirection.Forward);
        if (previous == null && next == null)
          return null; //can no longer move outward in either direction and text wasn t found

        if (previous != null)
          leftPos = previous;
        if (next != null)
          rightPos = next;

        var range = new TextRange(leftPos, rightPos);
        var offset = range.Text.IndexOf(findText, StringComparison.InvariantCultureIgnoreCase);
        if (offset < 0)
          continue; //text not found, continue to move outward

        //rtf has broken text indexes that often come up too low due to not considering hidden chars.  Increment up until we find the real position
        var findTextLower = findText.ToLower();
        var endOfDoc = textbox.Document.ContentEnd.GetNextInsertionPosition(LogicalDirection.Backward);
        for (var start = range.Start.GetPositionAtOffset(offset); start != endOfDoc; start = start.GetPositionAtOffset(1))
        {
          var result = new TextRange(start, start.GetPositionAtOffset(findText.Length));
          if (result.Text?.ToLower() == findTextLower)
          {
            return result;
          }
        }
      }
    }

If you want to highlight the match then it d be as simple as changing this method to void and doing this when you found the match:

textbox.Selection.Select(result.Start, result.End);