Mail, Contacts, and Calendar
Since they are related, I'll cover Mail, Contacts and Calendar first.
Let's start with Mail. In order to use Mail you need a me.com email address. If you are a MobileMe subscriber you already have a me.com address and you can use that one. You have to visit www.me.com/move to convert your MobileMe account to an iCloud account before using the new Mail (which basically looks like the Mail from MobileMe). Converting will move your calendar and address book also. Your Gallery and iDisk will still be accessible through the MobileMe site until June 2012.
The Mail app is very straight forward and works like you would expect any webmail interface to work. If you use an iPad it will look very familiar since it is modeled after it's Mail app.
To customize Mail you can go to the settings icon at the top right (looks like a gear) and set filters and preferences. If someone emails you who is not in your address book you can click and hold their email address from the address information and an Add Contact bubble will appear.
The Contacts app, like the Mail app, is pretty straight forward. Basically it is an address book on the Internet. It has a simple interface and you can switch between group view and list view by clicking the red bookmark at the top of the book. Group view lets you see your contacts organized by groups you may have set up (for example Family, Friends, and Work). List view shows all of your contacts in a list without any group organization. Just like Address Book on your Mac (which might be renamed Contacts in a future update?) you can add, delete, and edit contacts.
Finally, Calendar is basically iCal through the Internet so if you use iCal on your Mac it will be very familiar to you. You can add and edit events just like you can on your desktop. One nice addition to the Calendar app is to have multiple calendars. With multiple calendars you can seperate events for work and home or family and friends, etc. I use them constantly.
The best thing about all three of these apps is that they sync with any device you have running iCloud. If you add a contact through the web app it will be automatically downloaded to your iPhone or other iOS device. They will also be downloaded to your Mac that is iCloud enabled. The same is true for calendar events and when managing emails.