Hey There
For the most part, it is not a good idea to just use tags as a way to try and gain search traffic from them. They are possibly beneficial to users internal to your site. Users may read an article and want to read other similar articles, so having a few tags at the bottom of the post can be useful. Putting tags in your sidebar for navigation is rarely useful, but if done in a somewhat user friendly way it could work. I generally avoid "tag clouds" or having dozens of tag links in one spot.
In terms of the tag archives themselves (like mysite.com/blog/tag/tag-name/), tag archives rarely look different than posts themselves or other archives. Unless you have a giant site, with so many posts, and tags actually add a beneficial way to scroll through archives on a very specific topic, categories do this fine enough.
And for indexation - if it's a new site or a site that has NOT ever indexed tags I would advise to not index them moving forward either. Unless in a rare .5% of cases this is done in an extremely intentional and strategic way, not for SEO but for users and site architecture. (Think of a site like Smashing Magazine or Search Engine Land with LOTS of content, that's a rare edge case where using tags for navigation and architecture might make sense.)
If you HAVE indexed tags already I wrote an article on how to safely evaluate and noindex them.
In general, I would avoid tagging a post with more than 4-5 tags. Tags should always be different from categories (like more specific things).
-Dan