Ahh, tryouts! They're kind of like dating or a job interview, but with out the promise of romance or a paycheck!
First of all you should just chit-chat and see if you think you even like being around the person. Then dance and see if you think you and the person can work together. A friend of mine (quite a high-level competitor) gave me some very good advice once: he said that the ONE thing you can fix in a dance partnership is the actual dancing, so make sure you check out the other aspects of the partnership too.
Anyway, after you've danced a bit you can talk about goals...what competition would you shoot for, or would you decide to take lessons and practice together for a few months first and then pick a competition? What about schedules? How many days a week can you practice, and where? If you can't plan to get on a consistent schedule then it's going to be very difficult -- each partner has to make space in their life to cooperate with the other to make time for lessons and practicing.
You should talk about how many lessons to take, and from whom. One lesson a week is probably plenty for now -- you want to give yourselves time to practice what you are learning, of course. If you're not sure who you want to take lessons from, agree to try a short list of teachers and then pick together from there. It's important that both partners want to work with whoever will end up coaching them.
Try to get a sense of how the person feels about dancing and how it fits into their life. Sometimes if one person is really danced obsessed and the other is more "take it or leave it" then that can can cause friction. Most of all try to get a sense if you like being around that person enough to spend time and money on lessons and practicing. You don't have to adore the other person, but since most people dance competitively as a hobby it's better to find someone who you can work with at least on a "pleasant co-worker" level than to spend time with someone who annoys you too much.