The company suggested they'll either visit me or they'll want me to visit them as we view each other's products and discuss a partnership. If they want me to travel to see them, should I expect to be reimbursed? Who brings that up in conversation?
Only ask for reimbursement if you decide you don't want to work with them!
No - unless they volunteer it, it is your expense.
I would suggest you not ask for travel reimbursement - it raises a red flag on your financial resources. If down the road it becomes more of a workshop/training vs. the kind of meeting your describing, I would consider asking for the partner for reimbursement.
Having been, "the big company" for 20+ years, more than a red-flag for me, it is an immediate we don't want to have anything to do with these (little) guys. Partnership means that...each puts skin and resources, into the game.
OK.. I'm going to against the grain here and say that it depends on the purpose of travel. If this is a consulting type gig, then travel is billable, though managed by prior agreements and limits. If this is a technology development or sales partnership, then no, it would not be reasonable to ask for travel reimbursement.