We used Mailchimp, which has a great free plan but caps out at 2,000 subscribers. (To email 5,000 subscribers you end up paying $50/month.) We haven’t decided whether that’s the best route for us to take long-term, but Mailchimp provided developer-friendly forms that took minutes to add to our site.
We launched our company with a parody product.
Nicer Studio

I was in a similar place last year and went the Mailchimp route. 8k subscribers cost $75 a month. Instead I’d recommend using Amazon’s AWS SES service. The first 62,000 emails a month are free, and the rest cost just $0.10 per 1k emails. Pair that with a decent Wordpress-based Mail app and you’ll have a fine email marketing operating for 1/4 of the price.