How to make 2 tag feeds show up on 1 page?

Your best bet is to create a page template with the following code. Also you will need to decide if you will use a tag or category. The code below assumes you are using a category.

<div class="news">
<?php
  //The Query
  wp_query('showposts=5&category_name=News');

  //The Loop
  if ( have_posts() ) : while ( have_posts() ) : the_post();
  ?> 
    <h2><a href="https://wordpress.stackexchange.com/questions/2578/<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h2>
    <div class="post_content">
         <?php the_excerpt(); ?>    
    </div>
  <?
  endwhile; 
  endif;

  //Reset Query
  wp_reset_query();

  ?>
</div>

<div class="events">
  <?php 
    //The Query
    wp_query('showposts=5&category_name=Events');

    //The Loop
    if ( have_posts() ) : while ( have_posts() ) : the_post();
    ?> 
        <h2><a href="https://wordpress.stackexchange.com/questions/2578/<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h2>
        <div class="post_content">
             <?php the_excerpt(); ?>    
        </div>
    <?
    endwhile; 
    endif;

    //Reset Query
    wp_reset_query();
  ?>
</div>