Blocking doesn't alert the person you blocked them and for good reason, if they knew you blocked them they'd look for another way to harass you.
Most new phones can block a number, it's extremely easy to do, just need to tap the 3 dots to open settings when viewing a number and the option to block should be there. This will stop you from getting texts and diverts calls straight to voice mail.
Carrier level blocking can done too but may depend on what carrier you have. Verizon will block numbers, completely stops communication, they can't even leave a voicemail. Simply contact customer service, I'd assume most other carriers are the same.