I use MailChimp's STS - really just a wrapper on top of Amazon's SES. I might just use SES if I wasn't already running newsletters through MailChimp, although MailChimp's wrapper is nice.
The only event-driven emails I'm currently sending line up pretty well with Stripe's events, so I'm just detecting them when they're sent from Stripe to my server, and firing off emails when they arrive. But if I wanted to set up a drip campaign like you've suggested, I'd just set up a daily cron job.
Mailchimp is pretty good for this, and is very reasonably priced. Constant Contact is another often-cited service.
If you want to get your hands dirty with the scheduling, etc, you can use a provider like SendGrid.
I just SES with Cron Jobs. That way I have complete control of customization, and can optimize properly.
MailChimp. Two cents an email (one cent if you buy them in blocks of $1,000 or more), not exactly ideal in terms of customizability (corner me online if you want the full spiel) but easily adequate for the "baby's first drip campaign" use case. They're also super-sensitive about spam which is great for you as long as you're so squeaky clean the sight of your smile could blind people at midnight during an eclipse.