When I search for my website on Google, it comes up with the like to my site as "HOME" Instead of what I want it to be. Will it change once my adwords account is fully setup ? I have done all I can do on my end.
Usually it should show the title as your page title, if your page title is "Home" then you need to change that
and to add to that - your adwords account will have absolutely no effect on how your site appears in organic serps - it's completely separate, you're paying adwords for a sponsored listing, an advert. This will appear in addition to your organic listing.
You have not done all you can. For someone here to confirm the problem/solution we need your website name. . .
We would be able to help you better with a link to your website. Is your website a CMS? In that case, maybe you didn't update the title for your home page.
what ever the page title is given, it displays same in Sitelinks but when using more than one title, it chooses the best and displays as sitelink.