My app list on a new phone I would like to buy a new phone, and don't want to install my currently used apps one by one on it. Is there a way to automatically restore my apps on the new phone?
Sure is!  When you get your new android phone and sign in with your Google account, it will automatically start to re download your apps!  If you want to root the new phone, and the previous one (the one you own now) was also rooted, you could use titanium backup as well to back up your apps and reinstall the them with all their data on the new device.