I agree that Mailchimp is good and it's free for mailing lists with under 2000 subscribers. If you have a website or hosting package you can install a PHPMailer, although these normally are not as robust as Mailchimp, it would be free regarding ongoing costs and a good Web guy would be able to install one quite quickly so wouldn't cost too much to set up.
Remember, with all mailing lists, you have to make sure the recipients have opted in and there are now GDPR regulations regarding emailing recipients in 28 European countries to consider.
I hope this helps :)
Mailgun is great, as is MailChimp. MailChimp might be easier for you since it has email blast campaigns and templates, while MailGun does not.
I worked for an Email Service Provider for four years. There is a big difference between sending an email and having it get past the spam filter and inboxed.
Just like location is key in real-estate, deliverability is the key when it comes to email marketing.
If the prospect never sees the email - does it matter how cheaply it was sent?
I'd suggest a split test sending a segment of your via multiple senders, and then measuring unique opens & click-thrus.
If it is a purchased list - you'll want to scrub it to make sure your deliverability doesn't tank overnight if you send to a spam trap.
I'd suggest using 2 services at a minimum. 1 Service to send to cold email addresses, and when someone opens or clicks on a link transfer that email to the 2nd service. You may need to replace service 1 frequently if deliverability drops.
That will keep the openers and clickers from getting nuked when if you have a spam trap on your purchased list.
I would recommend checking out https://www.mautic.org/ (free - opensource marketing automation platform self hosted or on AWS) or https://sendy.co/ ($59 - one time fee) hosted on AWS.
The downside is that you will need to have someone build out the HTML email templates. Also, if you are not that technical, you will likely need someone to help you set up these platforms. I'd find someone on Upwork -- $15 to set up, and will definitely be more expensive for the email template.
The cheapest that I've found, almost like a light-weight Mailchimp, is EmailOctopus ( https://yourlink.to/email-octopus ) with a cost of $19pm for 10k subscribers. At this level of subscribers, you could also check out Mailchimp ($75pm), Campaign Monitor ($89pm), MailerLite ($35pm) or Get Response ($65pm). Hope this helps.
MailChimp is a good option but as stated earlier, you have to be mindful of emailing recipients in the EU because of the new GDPR (data protection) law. So whichever you use, make sure it's got an "unsubscribe" option.