I run a number of wordpress self hosted websites and have done for over two years, below you will find some tips and advice. If you ever want to hire me to help you with your wordpress website, feel free to send a message via the contact form.
Check web browser compatibility
I’ve run into problems with both templates and plugins with different browsers. The usual alignment and layout issues with Internet Explorer for instance. Worth checking your blog looks okay in all the popular web browsers. You don’t want to be sitting there using the Google Chrome browser not realising your blog is absolute disaster zone for IE users. If you don’t run multiple browsers you can use: http://browsershots.org/
Security Updates
An all too obvious tip here, but make sure you keep your blog secure with important updates. Many self hosted users let their wordpress installations fall quickly out of date, not helped by the automatic update not working for all, which is understandable considering the variables. Take the time to do your upgrades and protect your blog.
File Permissions
If you’ve been rather relaxed about your file permissions, worth doing the homework to evaluate what your file permissions should be. See Hardening WordPress for more information.
Backup!
There are various wordpress plugins that can help you backup your wordpress database and a regular ftp download wouldn’t go amiss either, to make sure your graphics and template mods are kept safe. Even if your not using a self hosted wordpress install, a regular backup of your posts and content is a good idea. A good plugin for self hosted websites is: WP-DB-Backup
Cache
If your blog is popular or you often have sudden peeks of web traffic, it is worth configuring or using a third party cache solution, which serves up static files rather querying the mysql server all the time. That way you keep your website loading times reasonable and at times functioning, instead of the site falling over. links: W3 Total Cache and WP Super Cache