Even standing under a lit porch light as I leave for work, I can make out the Big Dipper and Orion constellations. North star is easy to find, as is Arcturus, Vega, Pleiades, Pollux and Castor, Cassiopeia, Venus, Mars, Jupiter and Saturn are all easy for me to see with naked eye. Many more, just have to know where in the sky to look.
I've got a 4" refractor telescope and have seen some nebula, and the rings of Saturn. Andromeda was pretty amazing. Once again, it's knowing where to look
the main constellations are easy to see even from the cities. From a dark place, much more. The Pleiades, the Milky Way etc. i've never seen it as in the long exposures, but more like a band filled with incredibly many faint, tiny atars and a few big ones
With a 20 cm reflector, faintest I've seen in a clear night is Stephen's Quintet down at Mag 13+. With an UHC filter, lots of the large emission nebulae - the brighter ones like Cirrus are fairly spectacular, for the fainter ones the joy is in the hunt. A couple of quasars - here the joy is only in the hunt, they're pretty boring visually.
It's flat-out impossible that you see *nothing* but the full moon with a telescope - I can easily see the brighter stars through thin clouds with 99% light extinction with mine where I can't see anything with bare eyes any more.
If light pollution is very bad, consider using a filter which blocks the streetlight spectral lines.
Best bet for casual observing is a good pair of binoculars. Easy to carry around and they don't have to be expensive.
Also get a smartphone app like Google Sky or similar and your phone will point you towards the various objects in the sky. Before smartphones, it was way more difficult to do this; now it's so simple people take it for granted.
Well, here in Dallas, I've generally found that enough of the faint stuff is washed out by light pollution that observation with binoculars is less than easy: the average angular distance between the things you can see is greater than the width of the binocular's field of view by a fairly large margin, so random observation doesn't work: you basically have to be able to see something by naked eye to know where to point the binoculars.
The OP may be running into similar issues with his telescope: He's trying to observe randomly, and his FOV is too narrow relative to the angular distance between objects that aren't washed out.
For sure the full Moon was really blinding Thursday evening. It rose though the haze/pollution layer, looking enormous and yellow for more than an hour. I took my car and when I had it in my field of view, the reflected light was strong enough to dim everything else and I told myself "Hey, what's that ?? What object can fly, be round and emit so much light ?". Took 5 good seconds to understand it was the good ol' Moon.
Really, polluted atmosphere can be interesting... I mean, you don't need binoculars when the Moon is "naturally" magnified 1.5x or 2x ^^