Before Applying
Typically the two primary goals in visiting before applying are to gather extra information about the school and the area and to make a good impression on faculty and advisors.

Before hopping in the car or on a plane, it is important that you pick up the telephone and call the graduate advisor or faculty recruiter to set up an appointment. Specify that while you are there you would like to talk to at least one faculty member in the department to which you are applying. Your visit will be much more productive if you have an appointment. You will get to talk to the faculty, and they will not be frustrated by someone unexpectedly showing up on their doorstep and wanting time and attention.

You might also ask if they will reimburse you for travel expenses. This is a typical request, and they may be willing to help, particularly if they feel you are a strong candidate. Responses will vary: some schools will decline to give you anything while others may offer you a place to stay or some might even pay for everything if they have the money and feel you are an excellent candidate.

During Your Visit
Your primary goal for this visit is going to be talking to just about everyone who will talk to you.

Faculty
Your first priority should be to talk to whichever faculty members are available. This will provide you with an opportunity to interact with them and see if they are accessible and willing to talk to you. You can also take this time to ask questions. It is helpful if you have prepared questions in advance. When preparing, try to find questions to acquire information that is not already provided in the materials you have. This shows serious thought and your genuine interest in the program.

This opportunity also allows you to impress the faculty with your enthusiasm and dedication to your studies. You have a chance to set yourself apart in the eyes of the people you meet and possibly gain support for when your application is being reviewed. Take advantage of it! Be prepared to discuss your research interests, your past experience, your future plans, etc.

You also are going to want to talk to your Graduate Advisor. This person should have a wealth of information of the sort that you as a prospective graduate student would want. One of the things you want to make sure to discuss are your chances for financial aid. Often you will get a better sense of this from talking to the graduate advisor than you would from the books or written materials they send. (Which only makes sense since the books are written for a general audience, and the graduate advisor will be talking to you as an individual.)

Students
Your next mission will be to find other graduate students to talk to. Wander the building. Explore the labs. Search the library. Talk! Seriously, it is important to get as much information as possible, and talking to other graduate students may give you a fuller picture of the school than you will gain by talking only to faculty. Ask students about the professors: ask if they are accessible, ask if they are easy to work with, ask how long it takes people to finish and if there are certain professors with a reputation of making life difficult for their graduate students. Ask about the school, ask about the area, ask about the cost of living, etc.

The students will probably be more than willing to talk, but you must be careful to weigh all the opinions you receive and try to get multiple opinions on any given professor or topic.

Investigate
After everyone is sick of talking to you, you'll probably want to walk around and investigate your surroundings. Walk through the town/city/local community. Buy a newspaper and check the rental and real estate listings. Try to get a sense of the cost of living and what there is to do in the area. Take any free piece of information that isn't nailed down. Walk around the school campus. Check out the facilities (laboratories, library, etc.)

Overall try to get a sense of the place and decide if you would like to spend the time there that it will take to earn your degree.

After Applying, Before Decision
Visiting after you have submitted an application but before you have received a decision is similar to visiting before applying. Your purposes are often similar, but it could be a wasted trip if your application is rejected or if you are awarded little or no financial aid. However, it may still be worthwhile and help prepare you to make a decision about which school to attend when the responses start arriving. If you have the resources and time, it may be a nice way to maintain the momentum of your decision making process as you wait for your applications to be processed.

After Being Accepted
Typically the only reason that you would need to visit a school after being accepted would be if you cannot decide whether to take the offer or not. If you are having trouble deciding, visiting can sometimes help you reach a decision. Visiting after you have been accepted is generally a different experience than you would have had visiting earlier in the process. Why? Because at this point, the school/department has decided that it wants you. The faculty there will probably treat you slightly differently as a potential and desired student than they would as a future applicant.

Faculty
At this point in the process, you should know which faculty you might hope to work with if you decide to go to that school. Call ahead and ask the graduate advisor to set up appointments to speak with these people. See if they are accessible and willing to talk to students. Ask questions to help you differentiate the academics at each school you are considering.

Once on campus, you may also try talking to the graduate advisor to get a sense of where past graduates are now and what they are doing. This is not always easy to do since graduates are not known for keeping their former schools closely informed of their progress. Also, if the record is less than stellar, the graduate advisor will probably try to downplay or avoid discussing it in great detail.

Students
Talking to students can once again be valuable. If you walk into department offices, frequently there will be students there for one reason or another. You can introduce yourself as someone that may be studying there and ask all the types of questions that you would have asked earlier. However, at this point your perspective and questions will probably be much more focused on information that will help you to decide between one school and another.

Have fun with your visits no matter when you go and don't forget to set up appointments and be prepared!