Where Do I Put a Template To Add Using {{block type="core/template" template="template.phtml"}}

The template file is relative to your theme's template folder. But also taking into consideration the template fallback (see this for more details) you have 2 options:

1. add the template file in your current theme:

   app/design/frontend/[YOUR_PACKAGE]/[YOUR_THEME]/template/[CUSTOM_FOLDER]/[TEMPLATE].phtml
   

2. add the template file in the base package:

   app/design/frontend/base/default/template/[CUSTOM_FOLDER]/[TEMPLATE].phtml
   

Then, in your CMS page you will add {{block type="core/template" template="[CUSTOM_FOLDER]/[TEMPLATE].phtml"}}

Where [YOUR_PACKAGE] and [YOUR_THEME] are the currently used package and theme folders used on your store, and [CUSTOM_FOLDER] and [TEMPLATE] can be anything on your choice

Welcome to Magento SE.

An easy way to make this call from a static block is, add this to your static block:

{{block type="core/template" template="page/template.phtml"}}

Place page.phtml at this path, with your custom code:

app/design/frontend/package_name/theme_name/template/page/template.phtml

If you are still having problems, due to Magento's hierarchy, you can also place it here:

app/design/frontend/base/default/template/page/template.phtml

Be sure to flush your cache