Replace multiple strings with multiple other strings

Specific Solution

You can use a function to replace each one.

var str = "I have a cat, a dog, and a goat.";
var mapObj = {
   cat:"dog",
   dog:"goat",
   goat:"cat"
};
str = str.replace(/cat|dog|goat/gi, function(matched){
  return mapObj[matched];
});

jsfiddle example

Generalizing it

If you want to dynamically maintain the regex and just add future exchanges to the map, you can do this

new RegExp(Object.keys(mapObj).join("|"),"gi"); 

to generate the regex. So then it would look like this

var mapObj = {cat:"dog",dog:"goat",goat:"cat"};

var re = new RegExp(Object.keys(mapObj).join("|"),"gi");
str = str.replace(re, function(matched){
  return mapObj[matched];
});

And to add or change any more replacements you could just edit the map. 

fiddle with dynamic regex

Making it Reusable

If you want this to be a general pattern you could pull this out to a function like this

function replaceAll(str,mapObj){
    var re = new RegExp(Object.keys(mapObj).join("|"),"gi");

    return str.replace(re, function(matched){
        return mapObj[matched.toLowerCase()];
    });
}

So then you could just pass the str and a map of the replacements you want to the function and it would return the transformed string.

fiddle with function

To ensure Object.keys works in older browsers, add a polyfill eg from MDN or Es5.

As an answer to:

looking for an up-to-date answer

If you are using "words" as in your current example, you might extend the answer of Ben McCormick using a non capture group and add word boundaries \b at the left and at the right to prevent partial matches.

\b(?:cathy|cat|catch)\b

• \b A word boundary to prevent a partial match
• (?: Non capture group
  • cathy|cat|catch match one of the alternatives
• ) Close non capture group
• \b A word boundary to prevent a partial match

Example for the original question:

let str = "I have a cat, a dog, and a goat.";
const mapObj = {
  cat: "dog",
  dog: "goat",
  goat: "cat"
};
str = str.replace(/\b(?:cat|dog|goat)\b/gi, matched => mapObj[matched]);
console.log(str);

Example for the example in the comments that not seems to be working well:

let str = "I have a cat, a catch, and a cathy.";
const mapObj = {
  cathy: "cat",
  cat: "catch",
  catch: "cathy"

};
str = str.replace(/\b(?:cathy|cat|catch)\b/gi, matched => mapObj[matched]);
console.log(str);