Contacts:

The main instructions to get contacts from your phone to a sim card, memory card or usb device are here:

https://support.google.com/android/answer/168477?hl=en

If you want to backup the contacts database they should be in this folder on your phone or something similar to it depending on customisations made by the device manufacturer:
/data/data/android.providers.contacts/databases
You could just copy these files to your computer to backup the contacts database.
The easiest way going forward would be to add your contacts as Google contacts and these would sync between your phone and Google account automatically as long as you have your account sync turned on (settings ->  Accounts and sync -> Your Google account).

Files:

For most files, the easiest way I’ve found is to sign up for Dropbox account and load the app on your phone.  Photos you take can be automatically uploaded to your Dropbox and then the’re accessible on your computer like a regular folder.
If you do sign up for DropBox, do me a favour and use this link – http://db.tt/lpDpafmj I get 250mb extra storage for the referral and it costs you nothing.
You can copy files off your phone by simply plugging the phone into the computer using the usb cable for your phone and copy the files off as an Android Phone will show up in Windows as an extra drive.  Mac users will need to download Android File Transfer – http://www.android.com/filetransfer/.
Or if you want to do it over wifi and not need to plug your phone in, check out Airdroid or ES File Explorer from the app store.  Both of these apps make it easy to transfer files off your phone to your computer over Wifi.  Although ES File Explorer is more complex and aimed more at users who would know how to set up shared folders on their computer.

Paid Apps:

There are plenty of other apps in the Android Play store that will back up your data for you.  2 paid apps that I highly recommend are Lookup Pro which adds antivirus and security features as well as photo and contacts backup and Titanium backup for a more comprehensive backup of your entire phone although it requires your android phone to have root access.