My fiance is a UK citizen. I want to move there with my daughter. We are both US citizens. I understand I will need to apply for a join partner visa, which I need to apply for our daughter. Her father is a UK citizen and she is a US citizen. We plan to stay permanently. I want to apply for the right visas, so that we have fewer problems in the long run and make this as easy as possible. Thank you.
You stated in the comments that you plan to get married before applying for the visa, so you should apply for a family visa as a spouse for yourself, after you are married.
You also stated in the comments that your British fiance was born in the UK, and that he is the father of your daughter. This means that your daughter is a British citizen, and you should apply online for a British passport for your daughter, not a visa. Your daughter has right of abode in the UK, so I don't think she counts as a dependent for your spouse visa application fee, but I'm not sure.