The easiest thing to do would be to set up your static webpage and have all it's links point at the blog.  http://yourstaticpage.com and youstaticpage.com/blog

There is a way to write a new template page and have it become your home page. It's a bit complicated.

You need to create a new page template and upload it to the site.

Your new page.php must have this coding in it:

<?php

/*

Template Name: New-Template-Name-Here

*/

?>

where "New-Template-Name-Here" is replaced with the name of your page. Do not name it the same as any other page in your site.

After that, add all the usual HTML that creates the page you want. Upload the file.

Head over to your dashboard, go into Pages-Add New

Name your new page, but do not add ANYTHING in the body area.

Over on the right hand side, under Page Attributes, you will see a new drop down menu for Templates. Choose your new template that you just created and uploaded. Hit publish, then check your pages. You new page will be there.

To make it the "home" page, go into Settings –> Reading  and use the drop down menu to choose your new page as your static page.