Get a good file explorer like es file explorer. Then navigate folders to data>app and look for the app in the list. Long press it until options pop up. Tap copy, then navigate to a folder on your ext SD card and tap paste. Now the apk (application package) is saved to your card. Put that card in new phone and get file explorer and navigate to where you stored it and tap it. It should give you the option to install it.
This won't save the data for the app so if it's a game, progress will be lost. To save data you need to be rooted and get a backup app like titanium or similar.
Edit: Actually es file explorer has options to backup apps come to think of it. So it will probably be easier than my instructions.