Diablo II

Before You Play

Get the expansion, if you can. Comes with a whole extra act, new items throughout the whole game, two new classes, and several quality of life improvements, including a much larger item stash. Plus, you can get both it and the base game on the Blizzard website for $20.
Even if you only play single player, connect to the internet long enough to get updated to the current version (1.14 as of this writing).
The multiplayer community is pretty much dead, and some poorly-timed lag can easily get you killed, so stick to single player unless you're specifically playing with friends.

The best classes for a new player are Barbarian or Paladin, but all of them are totally playable.
For your first time through, you should be fine leveling up whatever skills seem interesting, just don't spread yourself too thin. If there's a late-game skill you want, hover over it to see which early skills give it bonuses. This should give you a pretty decent idea of how to allocate your skill points.
Strength and Dexterity are mostly important for meeting attribute requirements, but provide some damage bonuses. Melee should focus on Strength, Ranged should focus on Dexterity, shield users should pay attention to both. Vitality is good for everyone. Energy is probably your dump stat, since the game will happily drown you in mana potions and mana-boosting gear, if you want it to.
Azhara can reset your stats and skill points... once.

Buy a book of Town Portal scrolls as soon as you can easily afford it, and don't be afraid to use them regularly. 200 gold for a bit of TP is much better than a messy wipe.
If you don't know where to go to complete a quest, go to the last outdoor area you've discovered and hug the walls until you find a way into somewhere new.
Finding the teleport pad in each zone is much more important in multiplayer than single player, but even in SP, it's still worth tracking down just in case you get surprise-killed and have to run back to grab your gear.
Before you go into a zone/boss fight you think you might lose, open a town portal scroll but don't go through it. Then, if you die, you can come through from the other side, grab your gear, and run back out again.
You'll eventually get a quest reward where Charsi will imbue a non-magical item for you. The optimal thing to spend it on is probably a late-game piece of expansion-only headwear called a Circlet/Coronet, which have some crazy potential for what they can be enchanted with.

Make sure to stash your gold every time you return to town; you can spend gold directly from your stash.
Fairly early on, you will unlock a townsperson who can Identify gear for free. Once you get him, you can just warp back whenever your inventory is full rather than lugging around inventory scrolls.
Things that sell particularly well: Mana potions, throwing potions, Superior-quality armor (especially socketed), any magic gear that you identify because you might want to use it.
Better belts give more rows for your potion slots. Sashes/Light Belts give 1 extra row, Belts/Heavy Belts give 2, Plated Belts give 3.
If you're playing with the expansion, mercenaries can use gear you don't want/need, and ethereal equipment they're wielding doesn't degrade when they use it.
Never sell: Gems/Skulls, Runes, Rings/Amulets, or Rejuvenation Potions. About a third of the way through Act 2, you'll unlock a way to fuse three of one type into the next highest rank, (3 Chipped Ruby -> 1 Flawed Ruby, etc.) and Rejuvenation Potions can't be bought anywhere in-game and heal you instantly instead of over time.
Don't be shy about using gems/runes if you want to, just be aware that once they're in an item, they're in there for good. Save them for a Superior or magical socketed item, if you can.

Optimal play is to wait until your stash is full before you start using gambling as a money sink, but this is only really important once you get into the higher difficulties, and things like mercenary-reviving start getting crazy-expensive.
Gambling inventory is unique because you can immediately quit out of the conversation and back in to refresh the inventory. This can be used to easily 'farm' through several screens of stock to find better-quality base items. These tend to be worthwhile to gamble on even if they end up being stinkers just for the higher base stats.
Theoretically, you could sit refreshing Gheed's inventory for hours and get all kinds of way-out-of-depth gear to buy through gambling, but you might as well just spend that time playing the game.

Once you defeat the final boss of the last act, you will unlock what is essentially a New Game+ campaign difficulty called Nightmare. This includes new items, more powerful and varied unique enemies, and generally a much more difficult experience. If you beat it on Nightmare, you then unlock Hell, which is the same, but moreso.
Theoretically, a well-built character could get through Nightmare without a guide, but if you're going to go into Hell, or aren't 100% certain about your ability to take on Nightmare, you should look up a character-build guide and build one from the list that seems interesting to you, before going into Nightmare/Hell. Going into the new difficulty refreshes Azhara's reset, but doesn't let you keep one that you wasted.