Changing the text at the bottom of the page

The quick and dirty way is to modify the footer.php file in the theme. Looks like twentyten, which I believe includes the site name in the footer, which is why you’re having both change.

The code in the footer at around line 25:

<div id="site-info">
				<a href="<?php echo home_url( '/' ) ?>" title="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" rel="home">
					<?php bloginfo( 'name' ); ?>
				</a>
			</div><!-- #site-info -->

Make sure you do NOT delete the wp_footer function somewhere beneath this code, though, or you’ll likely break a bunch of plugins.