If you buy a new battery from AutoZone and the battery is easy to install for them and they don't have to remove a bunch of brackets etc to get to the battery then yes AutoZone will replace and install your new battery free of charge.
If you take the battery to a shop they might install it for around $10.00 or so but it can vary depending on the shop.
Even a handyman could install a battery in a vehicle but Autozone will typically install a new battery without extra cost when you buy a new battery from them.
I have customers come to me with a new battery they buy at places such as Walmart and I install it for $10.00 which is not all that much.
It doesn't usually take me more than 30 minutes or so to install a new battery for someone and I sometimes do at least a couple battery installs per day and other days I do other work on vehicles.