OK, so a backup and an iCloud sync are two different things.
When you did your Backup Now, it created an iCloud backup of anything you have selected on your device to be backed up. You can see what you have selected for backing up by going to Settings>iCloud>Storage>Manage Storage, then click on your device. You will see the Last Backup Date/Time, and below, you will see everything that you have selected to be in your backup. You cannot see the contents of a backup.
Contacts, Calendars, Notes, Photo Stream, etc. are not backed up to iCloud. They are sync'd between iCloud and your device real time. If you make a change to Contacts on your phone, you will see the change in Contacts on iCloud.com. If you make a change to Contacts on iCloud.com, you will see it on your phone.
So, again, under Settings>iCloud, since you already have Contacts and Reminders turned on, switch them off and then back on again to see if that will initiate the push. When you turn off Contacts, it will ask you if you want to keep them on your device. Say Yes. Then when you turn them back on, it will ask you if you want to Merge with iCloud. Say Yes. Once you have done that, check iCloud.com again. When you open the Contacts app there, you should see the same contacts there that you do on your phone.
Are you connected to Wifi? And when you say you have set up an iCloud account, do you mean you set it up under your existing Apple ID? If so, then are you signed on to iCloud on your device with that Apple ID, or with the @iCloud.com email address? If you are signed in with the @icloud.com email address on the device, try signing it out, then sign back in with the Apple ID that your iCloud account is under - Settings>iCloud, scroll to the bottom of the screen to sign out.
One last question - are you using iCloud on a PC or a Mac?
Cheers,
GB