The West is the largest region of North America and the period of what may be called "the old West" spans at least a few decades, so the answer to your question depends greatly on location and time. It is not possible to provide a simple, unified answer.

This is particularly true is we attempt to find a unified response of the "white" West to people of color, because how various people were viewed depended on the viewer and the viewed. The West was one of the most diverse places in North America when it comes to immigrants and people arriving from throughout the Eastern US. People brought attitudes and prejudices from their place of origin, but these attitudes were not necessarily shared by their neighbors. Because of the considerable mixture of people of diverse origins, the West tended to have a complex ethnic ladder that described who was better than whom, but each person's concept of the "ladder" could be different from everyone else's.

In general, the four "POC" groups can be regarded as African Americans, speakers of Spanish, Asians (usually Chinese), and American Indians. That is the order that one might encounter, but depending on the time and location, some might have placed the Asians lower than American Indians. Also, African Americans were sometimes regarded more positively, especially in pro-Union areas (which was a common Western attitude). This was helped by the fact that African Americans were relatively rare in the West, so they didn't threaten the employment of other people. And yet, in places were they were more common, African Americans could slip a rung on the ethnic ladder.

When it comes to the speakers of Spanish, the situation was made complex by the diverse number of places that these people called home. Californios could sometimes do well (but not always), and there were differences that separated the Chileans from Mexicans (and the other possibilities). Again, numbers and various factors determined how they were perceived.

Asians were widely scorned because they were perceived as willing to undercut the wages of others and they were numerous enough to be an effective economic force. The alien nature of their culture inspired a great deal of folklore about them - stories and beliefs that cast them in horrible light. Some came to their defense - Mark Twain wrote an eloquent defense of the Chinese immigrants as hard working and honest (even as he despised American Indians).

Because the ethnic/immigrant stew of the American West was so complex and simmered for so long, it is very difficult to offer generalizations.

Regarding saloons and places of business: the Chinese would not generally be welcomed in any establishment except their own (except when they were working in the saloon - then their labor was very welcomed). The intrepid sometimes went to Chinatown to sample food and drink or to buy exotic imports, and they were generally welcomed.

Larger communities often had saloons that catered to the larger ethnic/immigrant groups and these were sometimes hostile to outsiders: woe to the Cornishman who stepped into an Irish saloon! And yet, many other saloons tried to be open and welcomed a diverse clientele. Except that many states had laws prohibiting the sale of alcohol to American Indians, so we have to set that group aside.

There is a story that illustrates the complexity in just one time and place: After the passage of the 15th Amendment to the Constitution guaranteeing African American men the right to vote, a respected member of that community in Virginia City, Nevada stood in line to sign up to vote for the 1870 election. He signed his name on the list of voters, registering for the right, and the white person behind him refused to sign the list, saying he wouldn't put his name after a black man. One of the local newspapers describing the incident added that it was unfortunate that the two men had not stood in line in the reverse order because, the newspaper wrote, the African American was of such quality that he would not have seen it as a problem to place his name on the list following that of an inferior. So here we see the complexity of the situation: the new federal law granted the right to vote to a group within the minority; an African American man exercised his right and was allowed to do so without restriction; a white person expressed his racism in a way that was no doubt hurtful; the newspaper took the side of the African American and insulted the white man. So here we have a moment, which is in itself impossible to generalize: racism and support for an African American instantaneously expressed, making a generalization of that one moment impossible. The impossibility of finding a generalization in this case is multiplied by a factor in the millions if we are to consider the entire West from 1849 to 1900.