A search feature for fields? I've looked at the core search module, and from what I've been able to tell so far, it searches on the body of a node. 

What's the best way to get a search feature for a site that will search on specified fields of a content type?
You can use Views module http://drupal.org/project/views. 

With Views module, you will have a flexible method to pull content according fields of your content that you want to show, and with Exposed Filters feature of Views module (or with modules like Better Exposed Filters), you can search/refine by field of your content.

here is a documentation about Views: http://drupal.org/documentation/modules/views

here is an example: I want to filter and search by Title of my content. Then I can add to my View a filter to search by Title, then in Filter Criteria I can select Title and expose that filter to use, for example, a text field for search Titles:

this screen shot is about where is located the Filter Criteria area, and where is the option to add filters:



this screenshot is about where you can select if you want expose a text box to search/filter, in this example, by Title. The operator is very important to filter. In this case I will select Contains that seek a word that is part of title:



Also remember that you can add another filter, the filter by content type, to filter results by list of content type desired. This filter can be not exposed, or if you want, can be exposed to allow to users filter and search, according content types:



And finaly this screenshot have the View with a text field for filter and search by titles of my content:



I hope information be useful.