I've got nothing but my new scope and my eyeballs. My sky here at home is rather bright. What do I do?
If you can get to a dark sky where you can see the Milky Way, go! But it helps to learn how your scope works at home first.
Maps and Apps
I recommend getting an astronomy app for your phone, like SkySafari (the basic version is under $5). This will help you figure out where to point the scope or binoculars.
If you can't get a map or an app AND your in a bright urban/suburban sky, what do you do?
If you're also kind of new to this whole "the sky as a bunch of targets" idea, what do you do? Read below!
In spite all the hype about how high your scope can magnify, low power is your best friend. Scopes magnify, but they also brighten. Your telescope at its lowest power is a more delightful instrument than at it's highest (advertised) power. Cranking up the power requires some advanced skills: save that for later.
Telescope magnification is inversely proportional to the focal length of the eyepiece. The higher the number on the eyepiece, the lower the power. Find the lowest power eyepiece you have. This is your best eyepiece, at least for now. If you become committed to this hobby, you will upgrade to a good low power eyepiece and it will be your favorite eyepiece. So find that 25mm, 30mm, 32mm, 40mm, the highest number you have. Put that in the scope.
Why is low power good? It has a wide field of view, which means you'll be the mostly to see what your looking for. This is especially important for a non-motorized scope. And there are few things better than pushing a scope through the Milky Way or through the Virgo Galaxy clusters, or around the Veil Nebula or seeing multiple galaxies in the same field of view...
The moon is a great start. Even in daylight. If you can see it, point at it!
The sun is also a possibility, but ONLY IF YOU HAVE AN APPROPRIATE SOLAR FILTER (or SOLAR FILM). The SUN WILL INSTANTLY AND PERMANENTLY DAMAGE YOUR EYE IF YOU DO NOT HAVE APPROPRIATE FILTRATION!!
The moon has to be up. Like the sun, it's visible half of every day. Unlike the sun, its visibility changes (about 45 minutes later every day).
This can be done in the daytime.
Your scope probably has a finder scope on it. You'll want to make sure the finder is pointing in the same direction as the scope. There are adjustments and the finder. You can do this in the daylight or at night. Point the scope at a terrestrial target on the horizon (look through the eyepiece). Look through the finder. Adjust the finder screws until the crosshairs (or center) of the finder match the center through the eyepiece.
This can also be done in the daytime.
This sounds like a complete no-brainer, but the first time may be a little awkward with some scopes. Pointing an 8-inch dobsonian, for example, requires getting down low behind it and putting your eye near the tube to get a good angle. Schmidt-Cassegrains, because they're short, might require lining up along a mounting dovetail. If your scope is motorized then you'll have to get used to the controls. In fact, if you have a motorized/GoTo scope, save yourself some frustration: get the hang of things at home. Don't travel far without knowing your scope. Things are harder in the dark.
Video: Pointing a Scope
Your finder might make require some practice. Red-dot and straight-through finders are (literally) straightforward. Some optional finders have a diagonal on it so you can look "straight through". It also depends on your finder's field of view. It could be very narrow, requiring some good rough-pointing.
In any case, it's very easily-mastered skill. Practice at home.
Once you've got the scope eyeballed on a target, check the finder. If the target is in the finder, then move the scope until the target is in the center of the finder. Now, look through the eyepiece. Your target should be visible. If not, use a wider-field eyepiece - this is one with a higher number. Generally, it's a good idea as a beginner to work with the lowest power / widest eyepiece you have. As an expert, it's still a good idea to have a wide-field eyepiece ready to help find a dim target.
Video: Finding a Target
If the moon is visible it's a good target. It's a good target tomorrow, too. And the day after that. Every day the moon is different. The most interesting part of the moon is the terminator - the line/arc between light and dark. The craters and mountains on the moon stand in stark relief along the terminator and it's in a different place every day. Ironically, the Full Moon is the least interesting moon to look at: no shadows.
Outside of the Moon, there are just a handful of objects you can observe without knowing the sky at all. Inside of the Moon, it is completely dark.
Venus: Venus is the brightest object in the sky, behind only the Sun and the Moon. It's either in the eastern sky before dawn or the western sky after dusk. You'll see no detail except for it's phase: it has phases like the moon.
Jupiter: Jupiter is the next brightest. Jupiter is awesome. Binoculars will show the four moons that Galileo first spotted. Bigger scopes will show storm bands on the planet itself. Jupiter might be up at any time. It depends on the date.
Saturn: Saturn is very bright. If you know where to find it, it's easy to point a scope to. You can see the rings. You'll never forget the first time you see Saturn. It's dazzling. You're also likely to see its large moon Titan.
Mars: Mars is a good target for about one month every two years - if you have a big scope.
All these bright objects are visible from virtually everywhere.
If you're in a bright sky, you are limited to bright targets. Stars are bright, but are too small to resolve as anything other than white or off-white dots. They are interesting in pairs or clusters. Many observers love double stars - they vary in color and separation.
Star Clusters are very often visible in bright skies. The Pleiades Cluster (the Seven Sisters) is naked-eye visible and is fun to point a scope at. If you see a group of stars close together, point your scope at it.
If you know zero about the sky, but are able to get to a darker sky: go! If you can see the Milky Way, point a scope at it. Browse through it. Take your time. The Summer Milky Way, especially the Southern end of it, is a swatch of sky you can spend night after night enjoying for years: there's so much to see.
Okay, you have a map or an app. Now what?
You're limited by what stars you can see in your sky. Very bright skies show few stars. Stars are your guide to the good stuff.
Using a map or app is a matter of matching a bright star(s) or constellation on the map/app and the corresponding bright star(s) in the sky.
Which stars are which?
Asterisms (star groups) and Constellations help you identify the stars and identify regions of the sky. The Big Dipper (part of the constellation Ursa Major) is an asterism many people know. If you have a very dark sky, you can probably find the whirlpool galaxy with a sky map. In almost any sky, you should be able to find the Constellation Orion. You WILL be able to find the amazing and urban-visible Orion Nebula (it's the middle "star" of the three star "sword" below his very bright three-star belt).
One of the most valuable bits of knowledge you can have in observing is know the names of the three or four brightest stars in the sky at any given moment. From this you can know which constellations are up and very many maps/apps group targets by constellation.
Why Learn Constellations? So you know what those bright stars are. They're hard to tell apart by themselves and by their color. Knowing some guideposts, whether they are stars or constellations will help you synch up your map/app, your head, and the sky. It's how you get from the things your eye can see to what you need your scope to see.
Once you get an idea of what stars and constellations are visible, then you can "starhop" to your target. How To Starhop.
Once you starhop to you first "invisible" target, then you are officially an "amateur astronomer". Congratulations! Everything you do from this point forward is an extension of that skill you have: finding "invisible" heavenly treasures where most simply see a smattering (or a sea) of stars.
The "Permanent" Sky is slightly different every night (it shifts west 1-degree every day); every month the sky after sunset shifts west 15-degrees, bringing new targets into view. Note: if you're willing to pull an all-nighter you could see almost the entire sky visible from your location.
In addition to that fixed, permanent sky, you have the Moon and planets. Jupiter and Saturn are always great targets, they just rise very late at night (or early in the morning) for almost half the year. Again, if you're will to stay up or wake up, Jupiter and Saturn through good scope are amazing. Jupiter's moons will drag their shadows across the face of the planet. These are visible in modest scopes and are amazing.
So, how do you know what's happening?
Google "What's in the Sky?" You'll get a lot of results.
You can subscribe to newsletters and feeds alerting you to celestial events. There is always something interesting going on.
Subscribe to Astronomy Magazine or Sky & Telescope. They're wonderful resources, and your subscription keeps them funded.
Find a Local Astronomy Club. Star Parties are a great way to learn the sky and see and learn gear. Look for their outreach events.
Find an online forum. CloudyNights.com is the best and it's still free.