If you import all your contacts into gmail, you won't have to worry about them anymore, they will get synched to any new android phone that you buy. Any app you purchase in either the Amazon Appstore or the Android market will show up in your MyApps list and you can just redownload them. Free apps are not listed there, just take note of what you want to keep and redownload those as well. Your pictures and other media usually land up on the microSD card or you can move them their manually, or you can just save them on your computer. There are a bunch of backup programs available in the market that also allow you to save app data, I haven't tried any of them yet so I can't tell you which one is worth using, just read the reviews and go with the one you like best.
You don't need desktop software, that goes back to the days of Windows Mobile, or iOS where you had to backup to iTunes, until iOS5 came out.