My name is Evan, I’m a Guru here with Shopify.
Adding that Additional Details box is great option for your customers and help you get some valuable information. The ability to enable this section will depend on the theme that you're using. I don't see your store's domain name on this forum post, if you could paste the link here I'll be happy to take a look to confirm the steps for you.
In the meantime though, I'll be happy to add to give you some steps so that you can try it out yourself. The different options you have will depend on where you want the box to appear, and what it's for! If you want to have a Customizable product, like an engraving you would want to add this customization box. That will let your customer see the option on the product page so that they can add an engraving.
The second option is a little bit more general and will depend on the theme you use. Enabling Order notes will let your customer add information in the cart. This is handy for small messages, about the order.
Both of these options will appear in the Additional Details box in the order page. It just comes down to where you want them to appear. Implementing these will depend on your theme. If you have a third party theme that doesn't have these options then you'll want to look at having a Shopify Expert code it for you!
It's also worth mentioning an app like Form Builder that might be able to help you build something with more of automated, API integration!
If you have any other questions or concerns, please don't hesitate to let me know! I'll be more than happy to help!
Evan | Shopify Guru