First thing to do with brand new website?

It depends on what you want from your new website. If you want good traffic, then you monitor it, if you want something else, you monitor something else. It all depends on your wishes for new website.

Depends on what your goals are. What are your goals?

Here's what I do:

Niche Research - Look at the market, figure out if there is enough of that market even out there to warrent you busting your hump to get your hands on some of it. Then look at your competition and see what they are doing. Do it better.

Keyword Research - I learned through many years of free-diving with a spear gun to look for fish where the fish are. Not where I think the fish are, or where the fish should be. I fish where the fish are. Target keywords that the audience you want to reach is searching for, that is what they want to read about.

Unique and Quality Content - I can't over emphasize unique and quality content. This is where the real work comes into play. And trust me, there ain't no easy way to make this happen.

Analytics - Know where your fish are coming from, what they are looking at, and when and where they figure out they are better off looking elsewhere.

Webmasters Tools (google & bing) - All kinds of helpful info & tools to help you provide a better site for your visitors.

More unique quality content to back up your other unique quality content.

That's the first thing I do.

Write good content