My girlfriend and I are planning on visiting Yosemite (California) real soon. We plan on staying there for three nights and four days. My girlfriend and I are planning to visit Yosemite, California in the near future. We are planning to stay there for four days and three nights. four days and three nights (this means that you are leaving on day 4) four days and four nights (this means that you are leaving early on day 5)

As Doug and Thenuka have said, you can say either. In an informal situation, you can also simply say, "We plan on staying there for four days, three nights." You don't always need the "and" in daily conversation.

You don't need to say both. You can just say: We plan on staying there for three nights. or We plan on staying there for four days. However, it's not wrong to say: We plan on staying there for four days and three nights OR We plan on staying there for three days and four nights. Both are okay.