How can I change how my tag pages display?

First, the root of your issue is that WordPress has two very different template tags to output posts: the_content() and the_excerpt(). There are a lot of nuances with these and many get details wrong, I did my best to get everything right in post at my blog if you are interested that Make sense of WordPress excerpts and teasers.

In this specific case you want the_content(), but theme uses the_excerpt().

Simply editing it as in your follow-up in question is not really sloppy, actually it is the only (non-messy) way to change one of these to another.

Problem is you will lose your edits next time you update Twenty Ten (or WordPress, since Twenty Ten is part of package).

The slightly more complex, but solid and upgrade proof way is to create child theme for Twenty Ten and apply your tweaks there.

  1. Create child theme, see Child Themes in Codex. It uses Twenty Ten as example, should be easy to follow.
  2. Switch to your child theme, at start it would be identical to Twenty Ten.
  3. Create file loop-tag.php in your child theme’s folder.
  4. Copy content of Twenty Ten loop.php to that loop-tag.php you created.
  5. Apply your edits to loop-tag.php (you can probably just throw out extra stuff if you feel like it, this file will only load for tag archive pages).