Hi! What an exciting idea! It is very romantic to contemplate - and how delightful and rare to find partners who have the same goals and like each other so much that they would even want to do something like this. Really cool.
I agree with ClearPoint's advice that the smartest thing financially would really be to save up most of what you need before you go so that you don't need a loan. I'm also concerned about what will happen with your jobs while you are gone. It may be that you are professors and can go on sabbatical or perhaps are self-employed and can work on the road, but if not, I worry that any loan - borrowing on your home equity, getting a personal loan, or using credit cards - might be a problem if you end up with job uncertainty or no job when you return. You could return from your wonderful experience and have huge debt and no job.
Assuming that your jobs are all set, then I would think that setting up a home equity line of credit would give you the lowest interest rate. If you don't own a home, a personal loan from a bank would have a lower interest rate than credit card debt of 18% or more.
Another idea to raise cash might be to rent out your place for the 3 months you are gone. Be sure to put your car insurance in "storage" mode, too, if your insurance company does that (USAA does) to save there are well.
I am so excited for you both, and I so hope it works out. Best wishes and thanks for writing!