I have my own internal set of questions, based on the client and the type of video we're meeting about. Usually we chat by telephone first so we can have an idea of what the client is looking for. This helps us gauge if a client meeting is necessary and worthwhile for both parties. Most of our clients would not have an answer to:
Budget: (or if they do, it's often not realistic)
Greg is right - the first client meeting (and it could be done via phone or email, unless the meet and greet has not been done yet) should revolve around the big picture: what is the purpose of the video? What do you want the viewer to walk away with, or do? Will the video be a fixed length, or is it content based, in which case it could be any length, based on the story which must be told? Do you envision this as all graphics, graphics and video, stock video? Will there be talent and location shooting - if so, how many locations? And on and on...
It is smart not to assume much of anything going into the first meeting, but rather to remember that your first job is to solve the client's communication problem. Often that translates to making a video, but not always, and the 'type" of video also could vary quite a bit. Don't approach this like a guy selling "one size fits-all" solutions, because that risks you being dealt with on a strictly commodity-pricing basis. That's bad for you, bad for us.
The short list of questions you bring with you HAS to include some talk of the budget, but you don't commit to any figures in the first meeting. What you need to get out of them, after their list of needs, is how much they have available to throw at the project. What happens next is usually a variation of the old "I'll show you mine, if you show me yours' kind of game-playing. This is what they consider "negotiating".
What can help get past this stage into a frank discussion is laying out the steps of a standard production process, describing it in terms of days and day rates, as well as additional costs for out of the norm items. To make any kind of meaningful estimate, you'd need to see a script or at least a creative treatment document, and a list of the "deliverables".
Based on what is described in that document, you can figure the days, number of staff, crew, talent, effects, etc. Some people use the AICP form mentioned up-thread as a worksheet for making these calculations. THEN you can make a bid with some confidence.
Anything less than this process, you're really just guessing numbers out of your hat, and you could be too low or too high, but rarely just right.