What's a good image module?

...but first, you'll still need an image module - there are two basic standards:

Image - this is the "old school" node module for handling images as nodes. It is rock solid at what it does and is supported by many other modules, and comes with a variety of contrib. modules including an image gallery.

imagefield - this is the "new way" to handle content - using CCK. An imagefield allows you more flexibility because you can add an image(s) to ANY node type. There is also wide module support for imagefield.

the ability to directly edit the code that displays the image

This is not really the "drupal way". You need to think in terms of theming output rather than editing code. Drupal will produce output with css selectors - you simply write the css or modify the template to theme that output.

What imagecache will do for you is to automate a bunch of tedious image processing work - it will take care of scaling your images to various sizes for different uses (e.g, thumbnail vs. original)

That should get you going. My advice: do some reading in the handbooks about CCK, imagefield, and imagecache, then play with it for a while before trying to build something production quality - it'll be much easier once you get your head around how these things work together. If you are new to web development, stick with Image module - it is easier to understand, simpler to configure and use, and better supported in the documentation and forums at this point.

good luck