You can quickly jump to the next or previous moon phase even in the TPE web app using the previous/next event buttons:
When "next event" is clicked, the app will change the selected date and time to the next upcoming event. These include:
In this example, I've clicked two events forward to the next full moon:
Note: typically you don't actually want to be photographing the moon at the instant of full moon. And you won't even be able see it at the instant of new moon (it's not illuminated, by definition!).
Full moons are usually best photographed to the east at moonrise the night before the official event, or to the west the morning after, but check in Photo Ephemeris as times relative to sunrise/sunset and the directions vary significantly from place to place and month to month.
New moons are typically best shot to the west just after sunset, 1-3 days after new moon. The TPE crescent moon visibility indicator will tell when it is and is not visible:
The visibility is indicated by an icon and a rating A-F, according to the Yallop method:
|A||Easily visible to the unaided eye|
|B||Visible under perfect atmospheric conditions|
|C||May need optical aid to find the thin crescent moon before it can be seen with the unaided eye|
|D||Can only be seen with binoculars or a telescope|
|E||Below the normal limit for detection with a telescope|
|F||Not visible. Below the Danjon limit|