You can enable Custom Shipping Carrier(CCS) with Basic/Shopify plan as well but it required annual plan. Means you need to pay $(29*12) to Shopify.
After that you can contact either live chat https://help.shopify.com/en/questions#/contact/chat or email to Shopify for enable CCS.
You can customize in the app Shipping Rates by ZipCode based on zip-code as well as weight or price or quantity of the cart and product based shipping as well. Let me know if you need more help to setup the app.