Is it possible to jump back into MTG after 20 years away?
You can always play, but it will have some cost.
The rules have been neatened up and there are some core differences, but it remains mostly the same game.
Your existing cards can be played with, regardless of how good they are in light of current cards. You can make a few roughly even decks and play them against each other. The main risk of relying too much on old cards is that they can be confusing when teaching new players; a lot of them have had their text updated so you'd need to refer to Gatherer (e.g. Arcum&#39;s Whistle. Also, some of them don't present the stronger colour identities that help define Magic today.
If you just want to play together and get used to the game: I highly recommend building a few weaker decks out of a small-ish pool of cards. I did this on my return to Magic with a little more than the cards from 2 prereleases (About 20 packs in total), resulting in 4-5 decks which were varied but all playable against each other. Another way to do this is to ask for welcome decks at a game store, which are free 30-card introductory decks. Another option is to buy duel decks, which are full 60-card decks designed to play fairly evenly against each other.
You can also use your older cards in Commander decks (known as Elder Dragon Highlander), but these tend to require a larger card pool to put something decent together. Commander decks run 100 unique cards (unique apart from basic lands) with one legendary creature set aside that you can always cast. The format is designed for multiplayer, but there are more competitive two-player versions. Commander isn't pushed as a competitive format by Wizards, so different groups have very different philosophies to deckbuilding. Commander can be fun for new players, but there need to be enough decks and ideally enough players.
Constructed Formats
The following formats are competitive formats in which you bring a prepared deck. There are also banned cards which have been individually banned to prevent them from warping the game environment due to their strength (or for practical purposes).
Legacy and Vintage are competitive formats that use older cards, but far fewer players take part in these, and decks are streamlined and [incredibly] expensive. Games are often ruthlessly quick, as the best cards and combinations from all time are used. Legacy has a banned list, while Vintage allows 1 of each card on the Legacy banned list. Due to the complexity of the interactions and older cards, and the price, this isn't recommended for new or returning players.
Modern and Standard use more recent cards, and are better supported by Wizards and local game stores. Modern decks can contain cards that were printed in a Standard-legal set from Eighth Edition onward. Standard decks can contain cards in the 3 most recent blocks (each block having 2 sets). Due to the wide range of cards, Modern is a more stable format, with powerful and somewhat expensive decks. Standard is less expensive, and changes faster, but players need to change their decks as cards rotate out with the sets. Standard is okay for new/returning players, but still has a reasonable price tag.
Note that some cards from older sets may be legal if they've been reprinted since. Here are Ice Age cards that are Modern-legal, and here are Ice Age cards that are Standard-legal (currently only basic lands).
Limited Formats
While the constructed formats have varying price tags to be competitive, there are also limited formats, in which you use sealed booster packs of cards and build the deck before you play with it. Limited takes away the price factor in winning and distributes it between skill and luck. While each game of limited is cheap, you'll be using new packs each time you play. It can be a good way to play as you build up a base of cards for general play, or you can sell the cards after you play to minimise the cost as a hobby.
In Sealed, you build a 40-card deck using six booster packs and basic lands. This is a fairly quick way to play, and isn't too difficult to take part in once you're comfortable in Magic. Prerelease events use this format, and these are more casual events which are suited to newer players.
In draft, you pass booster packs around, picking a card each time. You then build a 40-card deck using the card pool you picked and basic lands. There are also 2-player draft formats which can be strategic and interesting. I've greatly enjoyed drafting with friends, although part of its success is that sometimes I've bought the packs and kept the cards (so newer players can experience the format without buying in). I love draft personally as it forces you to judge the value of cards that would be overlooked in constructed formats, and lets you build a variety of decks. However, I also don't mind the cost, and the cards go into my collection.
Finally, you have a combination of the two: Cube. This is essentially a constructed draft pool. You pick a large set of cards, and then take a random portion of it to play limited with. Some people spend a long time putting together and curating their cubes to make them interesting. You can also throw together a handful of cards you had otherwise.
Leagues
There's also the new format hosted at stores, Magic Leagues. This is kind of like an ongoing mini-sealed format. You use 3 packs and build a 30 card deck. As the event goes on over several weeks, you can add more packs to your card pool, and adjust your deck. This is designed to be a quick-to-play format which gives you experience building and then refining a deck from a limited pool. I haven't tried it yet myself.