How Lizardmen view themselves relative to the other races

"Me", A Name I call Myself

"Lizardmen" is a term given to the race by humans. Most Lizardmen are not even aware of the term and of those that are, they are often mildly offended by it.
Lizardmen actually call themselves "the First Children of the Old Ones." or "the First". For short.
In my own writing I often use "Lizardmen" for clarity because that's what Warhammer fans are familar with. but I often call them "the First" too. I more or less use these terms interchangeably.
The term "First" refers to all Lizardmen including, the Slann, Sauri, Skinks, and Chameleon Skinks. Any sapient Lizardmen.
The term does not apply to Stegadons, Carnosaurs or any other beasts the Lizardmen use in labor and war. These are "beasts". That doesn't mean the First don't respect beasts on an individual level or on an abstract love of nature level, but they don't consider them people.

Lizardmen and the other so called "Forces of Order"

The Lizardmen understand that they were not the only sapient beings created by the Old Ones and the Old Ones would not have created these races if they did not have an important purpose of some sort.
Many Lizardmen collectively refer to elves, dwarves, humans, ogres, and halflings as "The Prodigal Races." or simple "the Prodigals". They were created by the Old Ones to serve a greater purpose, but they clearly failed the Old One's plans.
Some Lizardmen distinguish between Prodigals and Fallen. The "Fallen" are Prodigals that didn't just stumble and fail, but they fell to dark powers and betrayed the Old One's vision entirely. This includes the Warriors of Chaos, Dark Elves, and Chaos Dwarves. Perhaps the Vampire Counts and Tomb Kings of Nehekara.
Just as the Lizardmen are "The First Children of the Old Ones" by a similar logic, elves are commonly referred to as "The Second Race." Some Lizardmen make a distinction between Fallen elves and other elves. Others cannot or willnot. It is the case that nine times out of ten, if a Lizardmen encounters an elven expedition to Lustria, that expedition is Dark Elves. From Skink to Slann, this has colored the First's attitudes towards elves in general.
According to published works, elves are called "Itz’xa’khanx" in Saurian meaning "Ringed by Ocean" referring to thier homeland of Ulthuan which is a mostly closed circular island in the ocean, but this is so clumsy to English language readers I prefer "Second Race",

Dwarves are called the Third Race. According to published army books, they are called Dro’ka’khanx meaning “those who delve.” It makes since to me that they might be colloquially called "delvers"
Dwarves don't invade Lustria very often so most Lizardmen are neutral towards them but when dwarves do make it to Lustria, they are often among the most relentlessly avarice driven. Most First are not aware of Chaos Dwarves at all.

Humans are called the Fourth Race. According to published army books, they are called "Xho’za’khanx "meaning “the untamed ones". I like "Untamed ones" but not the mess of apostrophes and xs. Others might simply call them "fools", or "mayflies" because they seem to die faster than the other races.
In recent times, the majority of treasure seeking expeditions are dominated by humans. It is relatively uncommon for Lizardmen to meet the Warriors of Chaos on any field but it is always memorable when this happens. It has also been noticed that the human race seems to create far more undead than the other races.

Ogres and halflings are collectively called "the Fifth Race" or somewhat more informally "the Eaters." To differentiate between them they are sometimes called Big Eaters and Little Eaters big Fifth Race and little Fifth Race.
Lizardmen rarely see either of the two Eaters, so few have strong preexisting opinions on them. Usually when they do see them, they are a small mercenary detachment to another force.

Outside of these races, races such as orcs, goblins, skaven and the like don't get a number because their races were not sanctioned by the Old Ones.

The things that must not be

Lizardmen, at least in my universe, refer to the Daemons of Chaos as "The Old Foes". While Lizardmen do not encounter Daemons very often in recent times they have not forgotten the Great Cataclysm and how the First Children of the Old Ones were the Thin Blue and Green Line which saved the world.
Even the Skinks, Sauri, and maybe the Kroxigor are aware that the Daemons are not natural life forms and they will keep coming back and this should not be.
All other enemies, or at least most other enemies, are mortal. These are typically ed the Anathema, creatures whose very existance is an affront to the Great Plan of the Old Ones.
This includes but is not limited to Skaven, Orcs, Goblins, Trolls, Fimir, Beasts of Chaos, and arguably the undead.
None of the First will argue that these creatures should not be exterminated but the Slann argue among themselves about the details. What order should they be exterminated and in with what methods. This is a major piece of the Great Debate. Merestar's Coalition believes the Fallen should be the priority targets because once the Fallen are eliminated, the First and the Prodigals can fight the Anathema together. Mazdamundi's Coalition barely acknowledges the distinction between "Prodigals" and "Fallen" and take a more literalist view of the Great Plan. What should not, must not be.
Orcs and Goblins are a special case. Through the magical scrying, the Slann have noticed the greenskins inflict grievous losses on the minions of Chaos and some wonder if there is a way to harness them or Anathema in general for the greater good.

Very little of the Slann's debating has trickled down to the Skinks and Sauri. Without the Slann telling them otherwise, the Skinks and Sauri will generally drop whatever they are doing to pursue the Skaven given the slightest opportunity, especially areas where the Cult of Sotek is strong. Unless they are general warmongers, most Skinks and Sauri will not go out of their way to chase after Beastmen or greenskins if these races are not directly interfering with the First's holdings or the Slann are not explicitly telling them to go after Anathema.
For their part, the Slann (when they bother to make distinctions between Anathema at all), make the Beasts of Chaos priority targets. It is the "of Chaos" part that sticks in their froggy craws. The Slann view the Beasts of Chaos as direct minions of "the Old Foes". Even the Slann that believe some Anathema can be redeemed or at least redirected towards greater ends do not believe the Beasts of Chaos can be salvaged in any way, and to the Slann's credit, Lustria is relatively Beastmen-free, especially compared to across the World Pond.