Sending them a thank-you note afterwards is a personal courtesy. Talk to many residents and faculty in the Department, as each will have a different perspective based on where they trained, their specialty, etc. Also, try to get some idea how many places you should visit, based on geographic constraints, competitiveness, etc. You can apply to many places than you ultimately decide to visit. It is difficult to assess one s own competitiveness grades, boards, AOA, Dean s Letter, etc.
Most programs seem to place a lot of importance on letters of recommendation.
Additionally, the applicant pool may vary from year to year and from program to program. When you meet with Dr. Bellini, you should have a broad list of programs to discuss. She will advise you based on your competitiveness if your choices are appropriate. Many prominent programs now have multiple tracks. Applying to all tracks at a given place makes you look undecided and uncertain.