Some time ago I wrote about The Quick Case Close Button as one means of skipping the Close Case page. In fact, there's another way to close cases quickly: you can show closed case statuses directly in the Status field of Case. This is not enabled by default, so many people miss this option.
You'll find the options that enable this in Setup->Customize->Case->Support Settings, and there are two of them.
Show Closed Statuses in Case Status Field does exactly what its name implies: it allows you to show closed case statuses right in that Status field. This means you can actually create a case and close it in a single step. Once you have closed statuses in that Status field, you can hide the Close Case button on the Case detail page by editing the Case page layout.
However, the page layout editor does not allow you to control the buttons on the edit page, and so you may be left with one redundant button, namely the Save & Close button. Fortunately there's another option for that on the Support Settings page, namely the Hide Save & Close Button. When this is enabled, the Save & Close button disappears from the edit page also.
For many Salesforce.com customers, this little addition should reduce the number of clicks it takes for their agents to create and close a case.