GIMP Version: 2.8.4
Operating System: Windows
GIMP Experience: Beginner Level

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I want to update my sites header but every time I design one and put it up, it comes out looking blurry and or pixelated. I have tried adjusting the resolution and size but nothing seems to work. Any thoughts?

Several things can happen:

1. Your browser it not looking at the site at 100% zoom causing images to be resized on the fly by the browser: check your browser zoom setting.
2. Your image is included in a HTML that forces it so a height/width which is not its native size, causing it to be resized on the fly by your browser: either check the HTML source code and look for "<image src="your_image.png" width=xxxx height=yyyy>" and check the value or take a screenshot of the page and compare the size to the original image (assuming it's not a zoom problem). Even a few pixels missing or extra are enough to make the image look damaged...
3. Your image is re-compressed by the website to save bandwidth, sometimes even changing the format (from PNG to JPG for instance): open the website, right-click on the header image, and save it to your disk. Compare it with your original image. Is it the same file size? is it even the same format?

The website URL would be helpful sorting out your problem. As ofnuts suggested it's probably in the HTML.

Here is the link to my site. I eventually want to do a complete redo but for now I want to start with the header.

http://www.preciousdesignphotos.com/

I checked your site.
Image (1) PD_Header.png html code 583x877px, actual size 1024x1542px.
Image (2) master12.png html code 378x307px, actual size same.

Actual size and html code is not matching.
I don't know how Homestead SiteBuilder works but I suspect, the 974x220px image in the screen shot is a place holder, so It looks like image(s) total width should be around 974px.
You may also need to use CSS z-index?

Even at full size the lettering of the "Precious "PD" Design" logo is not very clean... Such lettering should be produced at the right size directly, text is very sensitive to scaling (more accurately, our eyes are very sensitive to defects in text, and scaling always introduces some).

I am still learning about web design so what is the best way for me to fix this?

I've checked the web site in Firefox and Chromium, it looks fine to me. As far as changing anything, as long as you are using Homestead SiteBuilder I would ask for their support. SiteBuilder type web sites are finicky to troubleshoot from the outside.

IMHO, If your not stuck with Homestead SiteBuilder and your a rookie with web sites, I would seriously have a look at switching to Weebly or Squarespace. The bonus is you can easily set up a mobile version of your web site with Squarespace or Weebly.