The wrong thing to do: "Ok thank you, I appreciate you, bye bye!" -end of thank you page-
What you should do:
1) Use the opportunity to get more followers.
Put in your social media icons on your thank you page for people to follow you!
Another thing that you can do is to personally get your new subscribers to email/ send a message to a potential friend to get them to sign up. If you don't ask, you'll never get! Recommending someone to your business will not be the top thing on your subscriber's minds. Put it out there for them.
2) Use the opportunity to make some money.
Present them with an (appropriate and good) affiliate offer, or one of your own products (Fondly called the One-Time-Offer for internet marketers)
3) Set expectations for your newsletter
In marketing, when a customer's experience > his expectations, you get a satisfied customer. If a customer's experience < his expectations, you get a dissatisfied customer.
It's a pretty no-brainer to frame things like what you'll be sending, why you're sending them, how and your style of emails and what makes your newsletter different from others. For instance, if you're sending offers, you can make a personal promise that you'll review all products/ get a discounted rate etc.
Do this right and you'll have subscribers loving and opening each and every email you send.
Hope this helps.