The best way to get into astronomy is by firstly acquiring a decent pair of binoculars. For astronomy I would recommend a pair of 10 x 50’s, giving a decent magnification coupled with a wide field of view.
When viewing the moon the best time to see deatil is when the moons phase is either a Waxing (growing to full), or Waning (shrinking from full), phase. At these stages near the terminator (where the light meets darkness), detail will be seen, craters and mountain ranges casting great shadows.
When the moon is full less detail can be seen however the view is just fantastic.
A good start to viewing the moon is to acquire a lunar map, either a hard copy or an app for phone or tablet. From here identify the great seas i.e. The Sea of Tranquillity and the great craters formed by major impact events in the distant past i.e. Copernicus (see above picture).
The next areas to identify are the Apollo landing sites:
Binoculars are also great at spotting the moons of Jupiter.
I fully recommend getting the app “Stellarium”, which I use all of the time. This will show where to locate Jupiter in the night sky, also depending on the time of viewing, where each moon will be located. The cloud bands will not be visible, but it is quite amazing to see these small moons in orbit around their Planet. A tripod would be a good acquisition at this stage to steady the set up.
Even with light polluted skies just looking at the Milky Way reveals hundreds of stars not visible to the naked eye!
So hopefully this has whet your appetite into viewing the moon through binoculars, just get a deck chair, sit out under the moon, planets, and stars and look at the wonders out there!