We've got a great document that goes over the ways of setting up shipping rates, and you can definitely customize one to have a cost of $0 for certain cart totals: https://docs.shopify.com/manual/settings/shipping/shipping-rates#manual-shipping-rates
You can also do discounts for free shipping, if that works better!
If you want to display is on your homepage, you can use some apps. There are many apps do this, both free and not, on Shopify App Store
We have some great apps that I think you can use for your store:
- Offer discounts when customers first visit your store, Better Coupon Box, this app ask your customers to like/follow/subcribe you in return
- Offer discounts when customers checkout, Chechout Boost App, this app can offer discounts/ free shipping/free gift to decrease abandon rates.
Both these 2 apps have high review and Better Coupon Box is wisely used