Loop posts only excluding first post

You don’t need separate queries, you can run multiple loops on the same query-

// output first post
if( have_posts() ){
    the_post();
    the_title();
}

// output the rest of the posts...
if( have_posts() ){
    while( have_posts() ){
        the_post();
        the_title();
    }
}

You can also use rewind_posts() to reset the current post back to 0, as well as manually set $wp_query->current_post to whatever index you want and start the loop there (note: the post counter starts at 0, not 1).

If you only want to style the first post on the first page and not subsequent pages, you can check if it’s not paged with ! is_paged()

if( ! is_paged() ){
    echo 'this will only output on the first page';
    if( have_posts() ){
        the_post();
        the_title();
    }
}