Show all posts of all categories but excluding a category on custom blog page with pagination of my theme

If you take a look at the category parameters of WP_Query, you’ll see that you can use either cat=-12 or category__not_in' => 12 to exclude a category with ID 12 in your custom query.

I just also want to point out a few things here. showposts is depreciated, you should be using posts_per_page. Also 'showposts=6' . '&paged='.$paged is wrong. You should just use & between your parameters. So you should do something like this 'posts_per_page=6&paged='.$paged

To conclude, to exclude the category ‘Tour’, which is what I presume from your code, you can do change this line

$wp_query->query('showposts=6' . '&paged='.$paged);

to

$wp_query->query('posts_per_page=6&cat=-' . $category_id . '&paged='.$paged);