Minecraft – How to 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.