I've read that using keywords in your anchor text for internal links is a good idea.

If this is correct then should I be using a keyword instead of the word 'home' for example on
every page that links back to my index.html page.

Also, my page banner links back to my index page. If I also add another link back to the index page
using a keyword as anchor text, will it get ignored by Google in favour of the banner image link?

You should use anchor text in both your backlinks and internal links.

As for the banner. Google will take into account the first link it find in the HTML code. If you place the text link above the banner code that it'll be counted, if it's below the banner code (for example at the bottom of the page) than the banner code will be counted.

I have been using 2 anchor texts pointing to my homepage and both words have improved a lot in the SERPS.

Is it true that Google only count the first link to your site? Is it to the domain or to the page? What does google do with the anchor text on the 2nd link?

Only from one page. If you have links on two different pages and they link to the same URL than they'll both be counted.