I'm going to be visiting China next year, along with my father. My parents are divorced, but my dad is now dating someone who will likely be coming with us. They aren't married, so I'm not sure what word to use to introduce them. If I say something like, "这位是我爸爸，和我爸爸的女朋友", would that cause confusion? Is there another word for this, or would something like this be uncommon in China?
Using archaic phrases like 红颜知己 or 忘年之交 is a bad suggestion, these phrases come straight from dated Chinese movies. No modern human talk like this in real life. It is like saying " She is my father's fair lady" or " she is my dad's dear confidant." in English.
If they are very close, saying she is 女朋友 (girl friend) is perfectly fine in today's open minded society.
The safest option is to call her your dad's 朋友 (friend). or 好朋友 (good friend) if you are not sure they are officially a couple or not.
女朋友 is girlfriend in English, but in Chinese, it also implies these two people that are dating, are also sexually interacting, regardless of their age. If that is the case, then you can introduce her as your dad's 女朋友. but if they are only dating, and not sexually interacting, then please use 红颜知己 which means a very beautiful woman that is getting to know your dad. Or more commonly, you can use 女伴，a female that is dating your dad. On the other hand, if your dad is dating a much younger woman, you need to introduce her as his 忘年之交, which means ageless true love, or people may think that she is into your dad's money.
Also in some situation, 女朋友 and 女伴 could both means bitch or hoe, so to be safe use 红颜知己 if they are similar age, and 忘年之交 if one is older.
This is an interesting question. I think the answer depends on who you will meet (a well-educated relative, a college classmate, or an aunt who lives in the backward countryside) or even depends on whether that girlfriend is a Chinese or a white woman. I think it even depends on the extent you are westernized. I think if I am introducing a relative (such as my 60-year old uncle)'s "girlfriend" in a similar situation like yours, I would say: 这是黄阿姨，我叔叔的朋友 (This is Aunt Huang, my uncle's friend). And let others guess their relationship (they may easily find out according to the circumstances).