Because we want our sites to render across all/most major browsers, one of the top two things I do is: first run a site through the W3C Validator ->
Your site came back with a lot of errors in coding and some warnings that need to be fixed.
One thing you can do to fix these in your code is run your site with "Show Source" and "Clean up Markup with HTML Tidy" checked. With Show Source you can jump to these areas after you run with this turned on to see what the explanation is for the errors and warnings so you get a better understanding as to what needs fixing. With Clean up Markup... you can also find the code that Tidy spits out and use this code in your markup in place of the error code in your markup (just be careful with any <div id...class=...> classes that show up in the new code - you shouldn't use those unless you define them elsewhere in your site - the markup without the classes will work just fine in most cases). You can use the link at the top of the page to go to "Tidy Source." Oy. I hope this made sense; it's hard to explain if one isn't familiar with coding.
The thing about having invalid code, etc. is that not all browsers are "intuitive" enough to render sites as you would like them to (IE is a prime example and the bane of my existence

, Firefox is more forgiving). Using W3C standards compliant code will alleviate this problem.
The second thing I do is run sites through
Browser Shots (choosing the browsers and OS's I want to see rendered). This will give you an idea as to how your site will look to others. There are other ways to do this, but this is the simplest. I have three different browsers installed on my computer (that doesn't account for different OS's though), but it does give me a general idea as I design a site. In the end I run the site through Browser Shots or a couple others to see the final outcome.
HTH a bit and didn't confuse you totally!
Great that you have your site up and running - IMHO, just some tweaks and you will be good to go.