Matthew,
Looking at your blog page, you should have an opt in form on the right bar and maybe something at the top also. The good thing about aweber is that it's very easy to set that up.
You can incentivize people to sign up by offering a short eBook that would be useful and interesting for your target market. Maybe 7 Ways to Increase Your Traffic
Re starting up, you should look at Neil Patel's blog -- Quicksprout. There's a lot of useful information there
A) Send one email to EVERYBODY (ongoing clients, past clients, contacts who subscribe, website visitors, blog subscribers) or customize my messages for different groups?
That depends on what you plan to send out. If you send out a general newsletter or a new blog post, then you can put them all on one list. If you have something specific that you want to say to ongoing and past clients for instance which could potentially increase your revenue, then you're better off segmenting them.
When you add emails to aweber, you should segment them anyway so you know what source they came from.
When you send out an email, you will see which emails engages which groups the most. You will probably find current clients to be the most engaged in terms of open rates and click throughs.
Also be mindful of the fact that aweber does require double opt in sometimes so if you add a person's email from the CoC meetings, they will be sent an email confirmation that will opt them into the list. You can turn it off but you also need to delete the email confirmation message which is on by default if you don't want them to receive the email.
In any case, strictly speaking, you're not suppose to add the people you met at the meeting to your email list as it may be considered spam. So it's better to write them an email and tell them about what you're doing first and mention all of the really interesting stuff that you'll send them by including your best posts or something.
http://ug.infusionsoft.com/article/AA-00352/152/Marketing/Email-Delivery-SPAM/What-is-SPAM.html
B) Put whatever I want to share on my blog and share it via social media AND send an email to my list inviting them to check out what I posted so I get the most bang for my time? (LINKS BABY!)
That's the best way to share your blog. You should also ask in your email for them to comment, share etc. People on your email list are much more likely to do these things than a casual visitor