How can I create a farm in the Nether?

According to the Minecraft Wiki

Trees grow normally, and their leaves take a dead-looking brownish color like in desert biomes.

So trees can be grown as normal (as long as you are careful about making sure that lava doesn't just burn them all down), however

There is no way to place water in the Nether in a Survival game

Placing water source blocks with the bucket causes the water to instantly evaporate (with a hissing sound). You can still grow wheat in the Nether, however it will grow very slowly without water (also see the Wheat page on the wiki).

This also makes growing sugar cane impossible, however I believe that growing Pumpkin and Melon is largely unaffected by this as water only affects the rate at which the stem grows (which can be sped up with bonemeal), not the rate at which the Melon and Pumpkin themselves spawn. (The wiki does mention that Pumpkin cannot be grown in The End, but doesn't mention the Nether. I believe that Melon should grown there successfully but I haven't verified this).

Finally

Overworld mobs like pigs and zombies do not randomly spawn in the Nether, but throwing chicken eggs and constructing golems works as normal

So you will be unable to breed cows or pigs for food without using mods, however you can breed chickens by throwing eggs.

Its also worth mentioning that Nether Wart only spawns in the Nether, but can be grown in the overworld if you take a block of Soul sand with you.

The Minecraft wiki states (I've tried only wheat and watermelon) that all farmland crops can be grown in the Nether (growth speed is decreased). The only thing you have to remember is that once you gather your crops you have to immediately replant them. Otherwise because of the missing water (as stated from others here you can't have water source blocks on the ground in the Nether needed to grow sugar cane and speed up the growth of other crops) the dirt will revert to its default state and you have to use a hoe to make it suitable for farming again.

Personally I think that watermelons are the best food source in the Nether if I'm down there for a long time and my baked potato supplies diminish. Take some dirt with you and put the blocks in a safe location in the Nether (a stone house will do the trick, since Ghats can't blow stone blocks up). Use a hoe to turn the dirt into farmland and plant the watermelon seeds. Use bonemeal to speed up the growing. The special thing about watermelons and pumpkins is that once you plant the seed the ground doesn't reverse to its default state (in which you can't plant seeds in it). Pumpkins and watermelons are the only ones that produce "fruit" at the same speed as in the Overworld (tried it myself!), since the only thing that takes longer to grow in the Nether is the stem. Unlike wheat for example, which has to be replanted after gathered, pumpkins/melons provide infinite food once you have fully grown their stems. Pumpkins however are only useful in the Nether if you want Jack'o'Lanterns. It is not possible to create the recently added Pumpkin pie recipe without sugar cane and sugar cane grows only next to water blocks.

Here is a great tutorial for a fully automated watermelon/pumpkin farm by monkeyfarm:

I have a couple of those in the Overworld and they work great. The only problem is that they require plenty of resources and such that can be found only in the Overworld (iron, stone, redstone, sand (for glass; not Nether sand!)). But once you have build and secured it in the Nether, it will give you PLENTY of food.

Trees are also not that difficult to grow. You need some light sources, a safe location (note that making a huge room for your trees doesn't mean Ghasts won't spawn in it so be careful and take extra care!), dirt and tree stems. Growth rate is again the same as in the Overworld! You can produce tools (axes, hoes, shovels etc.), charcoals, torches and so on so it's essential to have a decent tree farm in the Nether if you want to stay there for a longer period of time.

You can simply take cow on the lead, bring very close to the portal, and teleport yourself to the Nether. Cow should be teleported with you, but lead probably don't, so take minimum 2 leads per cow. I haven't tried with sheep or pigs, but it should work too.

Yes,you can grow crops in Nether(at least wheat). Crops can grow without water at a slow rate.(Note:You cannot place water in Nether,and farmland will transform into dirt if it has neither water nor crops in it(I am sure that it applies at least to wheat crops),according to the wiki. Trees can grow normally,but they can be set on fire by lava. Just make sure that you bring dirt and seeds/saplings. And now about the village,just build some houses with at least 5×5 space horizontally. (I'm not sure about the height,but for maximum succes chance,put 5 spaces vertically. Then,bring some villagers through the portal,either from a village or from curing a zombie to populate the village.

This is an updated answer for 1.16 Minecraft. Judging by the fact you are beginning on a multiplayer server, my assumption is that you are in the Overworld. To my knowledge, wheat, mushrooms, pumpkins, and watermelon can all grow in the Nether, just at slower rates. You need dirt for all of these of course. 1.16 has added Crimson and Warped Trees, so your wood needs are fulfilled, but you can still grow regular trees in the Nether. For native livestock, you can make a Hoglin farm, which is on the more dangerous side but still functional if you get enough Crimson Fungus and can either kill them from afar or take some hits.