WordPress is designed so that all you ever have to do is write - i.e. you don't have to be a web developer to use the platform while you're blogging. You add functionality through plug-ins, which are optional modules you download, install, activate, and configure within the WordPress GUI. No coding skills required.
Getting WordPress
You can get a free WordPress blog at WordPress.com. You might also be able to set one up on your own domain through your web host with a one-click-installation tool (this depends on your host). Finally, you can usually find a developer who will install WordPress on your hosted server for free (Here's an example of an Arizona developer who offers free WordPress installation.)
To install a syntax highlighter, you'll need to be running WordPress on your own server. The free WordPress.com setup doesn't allow you to install additional plug-ins. But setting up WordPress on your own server is fast, relatively easy, and as I mentioned before, very well supported by the community.
Installing a Plug-in
You can search for plug-ins and install them directly within the WordPress dashboard. Just go to your dashboard, click Plugins then click on the big Add New button at the top of the page. From this screen, you can browse available plug-ins by tag or search for a specific feature.
Once you've found a plug-in you like, click Install Now and follow the on-screen instructions to install and activate the new system on your site. Then you have new functionality set up, and you didn't have to touch any code!
Syntax Highlighting
I use a syntax highlighter on my own site for displaying PHP and JavaScript code. It's called Code Colorer, and it supports a wide variety of languages and structures - you can even use a custom theme to color/style your code snippets if you want. Here's an example on my site using PHP. Here's an example using JavaScript.
Other Resources
Here are some other places you can look for more information regarding getting a WordPress blog set up on your own system: