Last I checked, most of my non-land cards are illegal for tournament play.
You should look into
Legacy, it's the ideal format for using older cards in my opinion. It basically allows everything, except cards which would need to be restricted, and those are outright banned. It bans cards based on power level; basically, if a card would necessarily make every deck better, it is banned. A good example of this is Sol Ring (artifact, costs 1 colorless mana, tap to produce 2 colorless mana). Every card is either legal or banned, there is no restricted list.
The other format some people play with older cards is
Vintage, however, I dislike this format and wouldn't recommend it. It's a very inconsistent format, because instead of banning cards that are "too powerful/necessary", it restricts them to one per deck. The card given before, Sol Ring, is another good example of this, since you'd be hard-pressed to find a good Vintage deck that doesn't use it and couldn't benefit from it. This has also caused them to restrict a good number of cards which would have been fine on their own, and are legal in Legacy, because of their interactions with these powerful/necessary cards (for example, Burning Wish).
In practice, Legacy matches tend to be more consistent and more skill-based than Vintage, and additionally, more fun, because Vintage is basically "whose opening hand got that 1 card in their deck that's horribly broken and will let them win on the first turn".
Therefore, if you're going to play one of those formats, I'd definitely go with Legacy. However, if your preferences differ from mine, I tried to describe them well enough for you to make your own judgement call.
As for MTG Online, all my decks are Legacy legal, however most people online play Classic format instead. This is unfortunately just as broken as Vintage (actually a little more so), and is basically the online equivalent to Vintage, suffering from all of its problems. All the cards which are legal in Legacy are also legal in Classic, however Classic made all the powerful/necessary cards restricted to one per deck; additionally, it didn't restrict any of the cards which combo with those, like Burning Wish. You basically have a format which combines the banned/restricted lists of Legacy and Vintage, except it always picks the less restrictive of the two.
I will say one thing though, playing Legacy decks in Classic has forced me to really improve the quality of my decks. If you have bad land ratios, or your deck is inefficient in any way, you simply cannot win. People are running decks which, if left unchecked, can win consistently within the first two or three turns. There are even decks which can win on the very first turn, albeit a bit harder to make consistent.
Anyways, that should be enough information to choose a format, for you or anyone else looking to play with older cards.