I synced my iPhone to my laptop. Can I safely wipe it clean now? I'm planning to buy a new iPhone. I'm going to exchange my current one for the new one. Before I give the old phone to Sprint, I'd like to wipe it clean (by going to Settings > General > Reset, etc).

I synced the phone to my laptop using iTunes. Before doing so, under "Info" I selected "Sync address book contacts" and all the rest.

Will I be able to restore that backup to the new phone?

Overall, am I going about this the right way? Is there an easier way?
Yes, you can. 

As long as you backup everything to iTunes, you are safe to wipe it clean. 

When you connect your new iPhone to iTunes it will ask, "Do you want to load data from the currently saved backup?" or, "Do you want to set up as a new iPhone".

You can also synchronize your iPhone with iCloud, directly from your iPhone. On your new iPhone, during first setup, you can choose to restore from iCloud.

Make sure you have the latest iTunes installed nonetheless.