The internet has changed the rules for a lot of things to the power of 10, such as dating, buying groceries and ordering takeout. But it hasn't changed basic manners. Meaning, you still have to write thank-you notes.
A Facebook post or a Twitter @reply is not a sufficient acknowledgment of thanks. Neither is an Instagram post. Here's a series of great examples of what not to do, courtesy the great Kim Kardashian:
OK Kim, wow, nice gift. Cute Kenzo sweatshirt. Do they make it in my size? I'm sure Nori will love it. But an Instagram post with an X Pro II tint does not a proper thank you make. You should write a note addressed directly to the gift-giver. Even a thank you email would suffice. In fact, the internet makes the handwritten expression of gratitude all the more meaningful. People almost never write them, and they almost always should.
I know this is a tough pill to swallow, but do not follow the example set by Kim K when it comes to both sex tapes and general etiquette. It's not the sentiment that's the problem, or even the act itself. I often tweet or Facebook about a gift someone has given me. Hey look I made a pie from that cookbook you gave me. But that's only after the thank you note has been posted. The social media acknowledgment is an add-on, not a thanks unto itself.
I'm not your mother, but listen to me like I am for a sec. You may not like writing thank you notes, but you have to do it. Someone was thoughtful enough to give you a gift? Send them a little note. Otherwise, you are an ingrate. It doesn't even have to be long. So next time someone gives you a present or does something nice to you, pull out a pen, and pop a note in the mail.
FYI you will need a stamp [/stamp/, noun: a small adhesive piece of paper stuck to something to show that an amount of money has been paid, in particular a postage stamp].