In general, Night Elves prefer the company of their own race to others, staying solitary and aloof. They are reclusive and non-Kaldorei are not normally seen on Teldrassil. Some don't trust other races and believe peace can fall apart at any time. Even the newly arrived Worgen seem to stay on the outskirts of the city or in the wilderness.
Humans and Night Elves met ten years prior to the current timeline. They fought together in the defense of Hyjal. Now the Night Elves have joined the Alliance. Humans are perhaps the alliance race that has the most interaction with the Night Elves, even sharing a marine route. Up until recently, the Stormwind Park was a spot for the few Night Elves within the city to stay and rest. Now that it has been destroyed, the Night Elves are staying outside the city, near the docks.
Dwarves and Night Elves have occasional interactions. They met alongside the human forces during the Third War. The only permanent Night Elf settlement in the Eastern Kingdoms is found in Dwarf land, the Greenwarden's Grove. Dwarves are permitted to dig in Night Elf territory for artifacts and research.
Night Elf and Gnome relations are strained. Night Elves take issue with Gnomish technology and it is said that only in dire situations will the two races cooperate.
The Draenei are somewhat distant allies of the Night Elves. They work together in Forest Song, Ashenvale, but beyond that, there seems to be little interaction. Even after a few years, ambassadors still work in the temple. During the Alliance summit, Tyrande and Malfurion were surprised to see the Draenei in attendance, showing how much they were aware of the race's current situation.
The Night Elves went to the Eastern Kingdoms to save the Gilneans from destruction. As the Worgen were the result of their curse, the Night Elves gave the refugees a home within their city. Many willingly stay outside the city in the wilds to hunt.
The Orcs are the main enemy of the Night Elves in the current day. Since their incursion into the forests of Ashenvale, they have not stopped and had only pushed deeper and deeper as time went on. After the cataclysm, the Orcs pushed even further into the forest, slaying many Night Elves in the process. They have been described as monsters.
Trolls are one of the Night Elves' longest foes, going as far back as 10,000 years. However, the Darkspear were not the breed that was normally encountered since then. As they fight alongside the Orcs, they are considered enemies. It is entirely possible that because of the past crushing defeats the Night Elves have delivered to the Trolls they have a hard time taking them seriously. During the advance of Hellscream’s Horde the Darkspear Trolls allied with the Alliance forces to defeat a common foe, this was likely a tense situation for the Night Elves at the time but they operated in bulk alongside their old enemies.
Tauren used to live in relative seclusion with the Night Elves but without any hostility between the two Kalimdor natives. This led to the Archdruid Malfurion Stormrage teaching the Tauren their lost skill of druidism. Since then the Tauren have sided with the Horde in the war against the Alliance, making them enemies. Those of the Cenarion Circle work together readily but on the battlefield, the two races will fight. Strong friendships have been made with this race however, we see this in the bond between Tholo Whitehoof and Anren Shadowseeker.
The forsaken have little contact with the Night Elves, save for some Orc supervised operations in Kalimdor. However, simply because they are allied with the Orcs puts them at odds with the Night Elves. Furthermore, their destruction of life, nature, and the balance would instantly make them enemies, regardless of their tie to the Horde.
The exiled Highborne eventually evolved into the High Elves, which then became the Blood Elves. The Blood Elf practice of magic instantly puts them at odds with the Night Elves. Even prior to their joining the Horde, the Night Elves invaded Quel'Thalas and set up spying camps throughout the forest, making hostile attacks against the Blood Elves. On their side, the Blood Elves seemingly take glee in any chance to attack and humiliate the Night Elves.
Goblins have a nasty habit of deforestation, strip mining and creating pollution. Any Goblin organization that does this earns the ire of the Night Elves. Because both the Bilgewater and Venture Co care little for nature, they tend to be at war with the Night Elves. The Steamwheedle Cartel, however, seems to be slightly less destructive and remains a neutral party. Because of this, it is unknown how the Night Elves would view the varying factions within the cartels.
More than 12,000 years ago, when the Kaldorei were still stumbling upon the Well of Eternity the Mogu had already conquered a swath of territory and enslaved the ancient Pandaren in it. Despite their best attempts to prevent an uprising, the Mogu were eventually overthrown by a Pandaren rebellion. Therefore between 12,000 and 10,000 years ago Ancient Night Elves and Ancient Pandaren were neighbors. At present there is no canon evidence of their relations during that time, however, we can assume that their shared dislike for trolls made them preferred neighbors as opposed to the alternatives. In present times the Night Elves seem accepting of Pandaren. There was no evidence of Night Elf monks until the coming of the Tushui Pandaren from the Wandering Isle in recent times. Under modern Pandaren tutelage, the monks craft spread to the various races of Azeroth, including the Night Elves.