For myself, part of the first day is always logistical: going over names and trying to make notes on pronunciation, and going over the Course Information sheet and exam dates. I've made an effort in recent years to make my Course Info sheet as short as possible (ideally 2 pages) and I try to emphasize my own personal quirks (I'm a stickler for not accepting late homework, and for taking off points if it's not stapled). I also make an extra effort to be energetic and to appear very approachable, so that the students are as at ease as possible if they don't know me already.

Ideally I have some interesting activity on the first day as well, almost always in groups (even if the class isn't usually group work). In my liberal arts class, we often do some code-breaking exercises and the students really like that. In my History of Math class, I have them invent a number system using only the symbols - and | and explore how to add, multiply, etc. In a class like Calc I I'm more likely to do review: I have a worksheet that has several examples of functions (some algebraic, others in a graph or table) and I have them answer basic questions about them. Usually I'll ask the students in all these classes to finish the worksheets as homework or in the next class and we'll go over answers the next day, but I don't collect this initial assignments. I would shy away from exams, mostly because a lot of my students are nervous about math anyway and I want to make it as positive as it can be at the beginning, even as I'm telling them how hard I want them to work.

I used to spend part of the first day collecting information (name, phone, email, major, etc) on a 3x5 notecard, but last semester I switched to emailing the entire class a list of those questions [this was simple using Blackboard] and also what their favorite book was, and then as their first homework assignment they had to email me the answers before the next class. I had warned them about this, and everyone responded, and I liked it a lot better both because it saved time and because it made sure that everyone emailed me at least once. I did reply to every email, mostly briefly.

Beyond syllabus and course info, if the class has a prerequisite (which almost all math classes do), I give them a worksheet with review problems (finding slopes of lines in a calc class, for example) and have them work on it in groups. I used to play name games, but it takes a lot of time, and it never seemed to help me remember the names anyway.