How do I check my calendar and make an appointment with the person I am talking with on the phone?
You can create the event in the calendar app and selecting to save it to your Gmail calendar. When creating the entry, you can add people you want to invite by adding their email address.
Put the call on speaker, then press the home button to navigate to your calendar. Once you leave the calendar app, pull the notification bar down to access icons related to the phone call (green handset to return to the call, red 'end' to end the call).