Is is possible to loop through both a category group and tag group in a for loop?
Categories: blue, red, orange Tags: basketball
Loop displays: basketball, blue, orange, red.
Something like this may work for you (untested).
{% set categories = craft.categories.group('myCategoryGroup').find %}
{% set tags = craft.tags.group('myTagGroup').find %}
{% set elements = categories|merge(tags) %}
<ul>
{% for element in elements %}
<li>{{ element.title }}</li>
{% endfor %}
</ul>
Sorting them alphabetically is a little more difficult. For that you may want to try twig's sort filter or, failing that, the supersort or craft-twig-better-sort plugins.