Personally, this is where I'd just open Sequel Pro and run the database queries directly. But if you aren't use to working directly with the database tables or performing SQL queries directly, then ExpressionEngine provides a very simple and easy to use tool. But… PLEASE BACKUP YOUR DATABASE FIRST.
EE2 offers under the
Tools Menu, the Search and Replace Tool.
The type of find you are doing is relatively simple. So basically, you will open the tool: Place
Search For: <h2>
Replace With: <h3>
Then select which field that you want to do the find and replace on, from the database list
Then perform the search. Now remember you need to do this for opening and closing tags, so don't forget the
Search For: </h2>
Replace With: </h3>
But just do this for each tag you need to replace. In fact, this tool, is perfect for this type of modifications. But it is powerful. So, please backup your database first, in case you select the wrong field, modify a template instead of a field, etc.