When I applied to graduate school the first time, for my master's, I wasn't at all sure what I was doing. I only visited one school, but that seemed to make a difference. The day after I applied (in fact, around 8:00 that morning), the dept. chair phoned me and not only verbally accepted me into the program, but also offered me funding. The other three schools I applied to (and did not visit), accepted me, but no funding was forthcoming, and they were not nearly as enthusiastic about me.
The second time, for my Ph.D., I only applied to two schools. I visited one of those, and again, that seemed to help. I didn't meet with everyone I wanted to (it turned out they were on Christmas break and I hadn't realized that), but I met with the two most important professors to my interests.
Talk to Faculty and Students
From my experience, the best way to make sure you're a match with a potential graduate school is to first visit, so you can talk to professors AND to other grad students (I found this very helpful!!). Secondly, make sure there are faculty members there that share your research interests, if you know what those are from the outset. I didn't have that for my master's, and I felt like I was walking alone a lot of the time.
Getting into Graduate School: the Series
More Advice on the Application Process