Once again, we are in a mobile age so it makes sense to list Android apps for moon phase predictions and research. Check them out on the Google Play store here:
Apple also has apps.
Current Moon Phase Predictor
This page lets you view the current moon phase and / or add it to your own web page if you so desire. The module can also be embedded in a range of web pages and used in a variety of applications. Note that this may not work on mobile.
Moon Phase Calendar
This page lets you predict the moon phase for an entire month. Moon pases can be seen for one year into the past and three years into the future. If you are planning long range and want to know when the moon will be at or near full or the new moon, this chart will help you do so.