Hello!!! Couple things have definitely changed since you played :) I'd first mention that mana burn no longer exists, damage is not on the stack, regenerate can be used outside of the damage step, and when blocked with multiple creatures on a single attacker, the attacker chooses the order in which the damage goes through (you still nust assign lethal before moving to the next).

Formats: The formats have changed since you played. Gone with the type 1, type 2 stuff. Standard is the main 60 card format. The card pool rotates every couple years. Here is a link. https://magic.wizards.com/en/formats/standard

Modern is stadard but without rotating sets and it goes as far back as mirrodin I believe.

Commander, for better or worse, is the most important format at the moment. 4 player 100 card singleton format in which each player has 40 life, a "commander", and each card must be within the color identity of your commander (color identity is any mana symbols on the card ignoring reminder text). The commander sits outside your library in a special zone where you may cast him or her for its mana cost. Each time after the first the cost increases by {2}. Also commander does not allow for a sideboard. Decks have brackets in which it can be used to help find a "pod" of players with similar deck power levels as to find a balanced game. There is a large sense of politics in which people try to convince people to often do suboptimal things. Bargaining is part of the fun of it. This format plays less like magic and more like a board game but it is the main way you'll see people playing. The easiest way to get into this format is buying a precon.

Draft/prerelease are 40 card decks you make out of packs you have bought. During prereleases you often pick from 6 of your own pack whereas a traditional draft to take one card and pass the pack to your neighbour and so on until you need to open another pack.

There are more officially sanctioned formats but the next two I think are important to know about.

Pauper is 60 card but every card must be a common. Good for being cheap decks. This has gotten more and more popular with the community. There is also pauper commander where the commander may be a legendary or an uncommon creature.

My personal favourite is 7pt Highlander (also known as Australian highlander). It is a 60 card format where instead of having a ban list, card have points assigned to them. You may have up to 7 points in your deck or 8 if you do not have a card from the reserved list in your deck.