HDE and Monica have already written excellent answers addressing this issue. I would like to contribute my $0.019999 from a slightly different perspective.
While it is certainly important to leave a good first impression on someone who you asked to check out a site, the issue runs a little deeper. Someone who has been told a couple of sentences about the SE site will probably be dazed for a few seconds, but they will either figure it out after a bit or ask you what they are supposed to do there. This is still an unnecessary hassle, but at least it works somehow.
Now consider a different usecase: Let's say I have heard about Stack Overflow for the first time. I don't know what kind of site it is (discussion forum, technical news site, price comparison website, social network, financial advice column, etc.), but I am curious to check it out and I find this:

What does this tell me about the site?
- They want me to join them in building a learning community. (I wonder what that actually means?)
- There was apparently an updated Code of Conduct. (Uhm, alright, as a first time visitor to the site, what is the relevance of this to me?)
- Developers come here to learn, share knowledge and build their careers. (Ok, so it is some kind of a knowledge sharing forum, right?)
- Login via Google, Facebook, or Display Name, Email Address and Password. (Wait, I'm confused now. Why are you asking me to sign up before I can even see the site to decide if I want to sign up or not?)
- They offer some Business Solutions which will help me understand, engage and hire developers. (Ok, now I'm really confused. This site is a job board now?)
- Their site uses cookies to show me ads and job listings. (Alright, that sounds reasonable, but what's a Stack Overflow network?)
My overall impression at first glance is unless I sign up, I won't be able to access the site. This is extremely unfortunate because one of the great things about Stack Overflow is logging in is not mandatory to access the site. If a new user has to scroll to get to your actual content, then you have already lost them.
To address this problem, I propose that users who have not logged in be presented the following layout. This is roughly the same as the second screenshot in Monica's proposal, with a few improvements:
- More real estate used for the site's core content, which is Q&A. As a result, it is much clearer at first glance what the site is actually about.
- Advertising is retained, including the "Teams"and "Jobs".

My proposal would retain the first screenshot from Monica's answer, except for taking care of the above two improvements.
I would also consider trimming the "cookie overlay" to one line, possibly with a "Learn More" link, because the current three line description is way too long, especially for a first time user.
Once the user has clicked Sign Up, they can be shown the "hero" banner. After the user has actually logged in, then they can be shown the Highly Noteworthy Quality questions, Code of Conduct, etc.
For completeness, since the question was about a non-Stack Overflow site, and at least presently, some of those are handled differently, here's what I propose for those:

The tabs bar and the "Ask Question" button can moved in the same row as the site title. You will have to cut me some slack for those elements not lining up perfectly. I could have figured that out too with some effort, but it demonstrates the idea well enough in my opinion. The design team can take it further and fix it within minutes.
Ideally, the site title should be in the same place as Stack Overflow was in the previous screenshot, but I understand that the company wants to use that space for branding, which is reasonable.