I’m so glad I found this forum! Thank you so much for your answers, SpankMarvin and Mark/t31os.
At first, I tried using the array of arguments shown above. This resulted in giving me posts that were in the category or with the tags. (meaning I’d get posts that weren’t in the category, as long as they had the tag attached)
Next, I used SpanMarvin’s code, cutting out $args and this resulted in what I was looking for. The posts that were within the category AND had the specified tag attached.
<?php $bios = new WP_Query('category_name=category1&tag=tag1');
if ($bios->have_posts()) : ?>;
...