How to forward request from catch all Servlet filter to Java Servlet?

Question

I am using Tomcat 10. I have set up a catch-all servlet filter in a Java Web Project. The web.xml looks like this:

  <filter>
    <display-name>UrlFilter</display-name>
    <filter-name>UrlFilter</filter-name>
    <filter-class>com.example.UrlFilter</filter-class>
  </filter>
  <filter-mapping>
    <filter-name>UrlFilter</filter-name>
    <url-pattern>/*</url-pattern>
  </filter-mapping>
  
  <welcome-file-list>
    <welcome-file>index.jsp</welcome-file>
  </welcome-file-list>

Requests are getting passed to filter but I want to forward these requests to a Servlet and from the servlet pass data to JSP.

public void doFilter(ServletRequest request, ServletResponse response, FilterChain chain) throws IOException, ServletException {
        
        HttpServletRequest req = (HttpServletRequest) request;
        
        String path = req.getRequestURI().substring(req.getContextPath().length());
        String pathInfo = req.getPathInfo();
        
        if (path.startsWith("/resources")) 
        {
            chain.doFilter(request, response); // Goes to default servlet.
        }
        else 
        {
            //TODO: forward all non-static requests to Servlet
        }
    }

My Servlet

public class MainController extends HttpServlet {
    private static final long serialVersionUID = 1L;
       
    /**
     * @see HttpServlet#doGet(HttpServletRequest request, HttpServletResponse response)
     */
    protected void doGet(HttpServletRequest request, HttpServletResponse response) throws ServletException, IOException {

        String path = request.getRequestURI().substring(request.getContextPath().length());
        String pathInfo = request.getPathInfo();
        
        System.out.println("path: " + path); 
        System.out.println("pathinfo: " + pathInfo); 
        
        if (path != null && path.equals("/")) 
        {
            request.getRequestDispatcher("/pages/home.jsp").forward(request, response);
            //response.flushBuffer();
        }
        else 
        {
            request.setAttribute("keyword", path);
            request.getRequestDispatcher("/pages/results.jsp").forward(request, response);
            //response.flushBuffer();
        }

}
}

I have two questions:

1. How can I forward all requests from the doFilter method to Servlet?
2. How can the servlet then process different JSP templates for two URLs / and /*

I don't have the option to use a Java framework to do this and I am new to Java web development. Please correct me if I approaching this in a wrong way. Thanks in advance for your guidance.