Change color and font for some part of text in WPF C#

If you just want to do some quick coloring, the simplest solution may be to use the end of the RTB content as a Range and apply formatting to it. For example:

TextRange rangeOfText1 = new TextRange(richTextBox.Document.ContentEnd, richTextBox.Document.ContentEnd);
rangeOfText1.Text = "Text1 ";
rangeOfText1.ApplyPropertyValue(TextElement.ForegroundProperty, Brushes.Blue);
rangeOfText1.ApplyPropertyValue(TextElement.FontWeightProperty, FontWeights.Bold);

TextRange rangeOfWord = new TextRange(richTextBox.Document.ContentEnd, richTextBox.Document.ContentEnd);
rangeOfWord.Text = "word ";
rangeOfWord.ApplyPropertyValue(TextElement.ForegroundProperty, Brushes.Red);
rangeOfWord.ApplyPropertyValue(TextElement.FontWeightProperty, FontWeights.Regular);

TextRange rangeOfText2 = new TextRange(richTextBox.Document.ContentEnd, richTextBox.Document.ContentEnd);
rangeOfText2.Text = "Text2 ";
rangeOfText2.ApplyPropertyValue(TextElement.ForegroundProperty, Brushes.Blue);
rangeOfText2.ApplyPropertyValue(TextElement.FontWeightProperty, FontWeights.Bold);

If you are looking for a more advanced solution, I suggest reading this Microsoft Doc about Flow Document, as it gives you a great flexibility in formatting your text.

You can try this out.

public TestWindow()
{
    InitializeComponent();

    this.paragraph = new Paragraph();
    rich1.Document = new FlowDocument(paragraph);

    var from = "user1";
    var text = "chat message goes here";
    paragraph.Inlines.Add(new Bold(new Run(from + ": "))
    {
        Foreground = Brushes.Red
    });
    paragraph.Inlines.Add(text);
    paragraph.Inlines.Add(new LineBreak());
    this.DataContext = this;
}
private Paragraph paragraph;

Use the Document property of the RichTextBox.