Using a programming API with blockchain
blockchain.info API is not related to Bitcoin-core, you can send commands to Bitcoin-core without the need to blockchain.info

for example to get full details about non-wallet transaction enable txindex and call getrawtransaction for wallet transactions call gettransaction and so on.

You can find all RPC calls here and you can send RPC commands from PHP, Nodejs etc..

if you want to have the full blockchain synced locally you can use bitpay insight (open source blockchain explorer).

or simply as you suggested you can use blockchain.info API without installing Bitcoin-core.