H1 and H2 Tags in 2013 Theme SEO

Hi there.

I’ve recently built a site using the twenty thirteen theme. It’s at https://robinthepiper.co.uk

I’m concerned that from an SEO perspective, the default behaviour doesn’t make good use of the h1 and h2 tags.

I want to keep the appearance of the header the same, but use my “h1 and h2 SEO capital” elsewhere, with custom content, perhaps hidden with css.

Any help with how easiest to achieve this?

I could copy the styling for h1 and h2 and put it in a span in the header code, perhaps? I’m worried that I might mess things up if I try this, due to inherited css styles that I don’t fully understand etc.

What about my idea of hidden h1 and h2 tags in the main body with more SEO useful content – is this an acceptable thing to do ?

I’m aware I could make the header text into an image, but I quite like the way that the site displays on mobiles and don’t want to mess that up.

Any help much appreciated.