Lisa here from the Shopify Social Care Team.
If you're looking for a free solution for sending a customized thank you email, Happy Email allows you to easily do so. No matter how many emails you're sending, or how large your email list is, the app is free and sends an automated response. The design of this email is intended to look like a personal message to your customers:
You can create a discount code in your Shopify admin, and include this in your emails when you're customizing them. Here's how to create a discount code. Once you've created the discount, you can press "Promote" and get a shareable link that will automatically apply the discount to the cart when customers click on it from the email. Alternatively, if you decide to go with MailChimp, when making an email template there's an option to add a "Discount block" that allows you to add a Shopify discount code and track the data from it.
MailChimp and Klaviyo are comparable, with Klaviyo being on the more expensive side. They're both intended for email marketing campaigns and sending newsletters to your email list. Klaviyo is effective for larger businesses who want something extra, such as audience segregation based on purchases, and advanced reporting. MailChimp is great for new businesses, mostly because it offers all the features sufficient for startups, free for up to 2,000 subscribers. Therefore, it comes down to a personal business decision for which platform is best suited for what stage your store is at. Here's a recent comparison of Klaviyo and Mailchimp by Whole Design Studios.
Hope this helps with your decision for what thank you and email campaign provider you'll go with. If you have any questions along the way, feel free to reply here.
In that case, as a free alternative for thank you email's that allow you to use your logo, try Shopify's app called Kit. Learn more about Kit's thank you email's, and download Kit for free here. This app is primarily designed to assist with Facebook marketing, however, you can choose to just use the automated thank you email function.
Let me know how this goes. ?