Duplicate and rename these folders:

/skin/frontend/default/
/app/design/frontend/default/

Now upload your new folders:

/skin/frontend/newTheme/
/app/design/frontend/newTheme/

Our new package name is "NewTheme" and our theme will be called "default"

First, disable the Cache. System > Cache Management, select "Disable" fromthe All Cache dropdown and save.

Then go to System > Manage Stores

Click Create Website

Enter a name for your store. We recommend using your domain name. (NewOZmerceStore.com)
Enter a code for your store. We use the domain name without the .com (newozmercestore)
Click Save Website.

Click Create Store

 

Select your new Website from the dropdown (NewOZmerceStore.com)
Enter a name for your store. (NewOZmerceStore.com Main Store)
Select your Root Category

Click Create Store View

Store: select our new Store Name (NewOZmerceStore.com Default View)
Name: enter a unique name. (NewOZmerceStore Default)
Code: This unique code is what we reference in our index.php file. example: newozmercestoredefault
Status: Enable

Ok, our new store is setup! Now we need to apply our custom theme that we uploaded in Step 1.

Go to System > Configuration

In the top left you will see dropdown labeled "Current Configuration Scope". We need to select our Website Name
(NewOZmerceStore.com)

Under General, click Design

Under Package, uncheck "Use default" and type the name of your package into the text field. The package name in our case is NewTheme (See Step 1)
Under Themes, uncheck all "Use default" and type your theme name into the text field. The theme name in our case is just "default"

At this step, you could also customize your HTML Head, Header and Footer text.

Ok the new package and theme has been applied to our store. Now we need to tell index.php to use the new store we have setup.

Open index.php and edit this line (usually the very last line in the file):

Mage::run('default');
...to use the Code of your new Store View. In our case, we used the code "newozmercestoredefault" (See Step 5). Our index.php file then contains this:

Mage::run('newozmercestoredefault');>

All Done! Just refresh your home page and you should see the new theme.