I'm lucky enough to work as an email marketing consultant, so I get brought in to help others get buy-in for what they want to do with email.
Recently had a client who approached us to help them on several fronts in their email marketing as it was being restricted in favour of another team. Our first step was to produce a review for them as an outside authority.
Getting buy-in is often achieved by addressing the stakeholder's pain-points. They're thinking "what's in it for me?" or "so what?" - Use that to your advantage and support it where you can with numbers, because stakeholders love numbers.
Dropping in a little late now, and I hope you've been able to take in some of the comments of others before me - but I thought I'd drop my 2 cents in as well.
As you're just getting started, and by your own questions I'd recommend you start with MailChimp. They're particularly good for new and small companies who are just getting started with email marketing. They have some great resources that you can quickly read through to get your start up set up and running in no time.
Those email clients you listed (Gmail, Outlook and Apple Mail) are not designed to send out such bulk emails, and anecdotally I've heard that they cap ~40-50 recipients, but I have nothing hard to back that up.
Rather than buying a list, which everyone else has already spoken against, perhaps you an look at other avenues of promotion, ad space on related emails or other channels? Twitter cards? Facebook custom audiences? There are many ways that you can promote your business and build your list at the same time, and really you should want your list to be YOUR list!
All the best,
Do you have images turned off by default? This disables the promotions tab from showing the emails as the pinterest-esque cards.
source: I have a couple of Gmail accounts, one with images on and another with images off
I studied Performing Arts, but I've always been doing email marketing. Also, as far as I am aware, I am the worlds first second generation email marketer. #lameclaimtofame