Hi and welcome to MTG!
So, Magic is a game with many formats, similar to how Hearthstone has Standard and Wild modes. I'll list the most common ones for you:
Standard: Cards that were released within the last few "blocks". Right now this is every main set released until now since Battle for Zendikar, which came out October 2015. This format rotates cards out as they get older, kind of like Hearthstone's Standard.
Modern: Every card that's ever been legal in standard since the release of Eighth Edition, in 2003. Cards don't rotate out of this format, but if a card does bad things to the game, it can be banned from tournament play.
Legacy: Every card that's ever been printed, with some banned cards. Magic's been around since 1993, so that's 10 years worth of cards that aren't in Modern!
Vintage: Every card that's ever been printed, period. Nothing is banned in Vintage, although you are "restricted" on certain powerful cards. Normally you can play up to 4 of any given card in your deck, but restricted cards you can only have one in your deck.
These are the "core" formats, where you play with a typical 60+ card deck. Vintage and Legacy are almost certainly too expensive to recommend to a new player (competitive deck prices reach into the thousands), and even Modern is pretty pricey (competitive decks are minimum a couple of hundred dollars, and some are much more). Standard is not terribly expensive, but since cards "rotate" out of the format you won't get to keep playing the same deck forever.
If you want to start playing, these competitive formats aren't the easiest way. The simplest route is to buy starter decks and just jam a few games with your friends! It won't line up with the deck archetypes you listed, but you can move on to building those later, when you have more experience. In the most recent set released, Kaladesh, the starter decks are called "Planeswalker Decks". Buy one of each and start playing with a friend, just to get a better grip on the rules. If you want, you can also buy packs (from Kaladesh or from whatever set looks interesting to you) and add/replace cards of the appropriate colour to your decks, to customize them.
Once you've played for a little bit and have a good grasp on the rules, you can start buying singles to build a deck for whatever format you choose (as yes, that is far more efficient than cracking packs), or if you want to play at a game store/more competitively, you can start playing Limited. In Limited, you crack packs and then build a 40 card deck out of the cards you got and play against other people. It's much cheaper (at least in the short term) than any other competitive format, everyone's on an even playing field because you're all opening the same kind of pack, and it's a very interesting, different experience to playing with constructed decks!
I hope this was helpful and I hope you have fun!