I have a Celestron Astromaster 114 EQ, which I saw was highly recommended as a great beginner scope. But I just can't seem to find anything in the sky with it. Yes, I've removed the lens cap, yes I've tried aligning with Polaris. Unfortunately, I think it is not properly balanced, and I can't seem to get it to balance, and I am totally unsure if my finderscope is aligned properly with the viewer. How can I check that, by the way? Everytime I blink the finder scope seems to have moved slightly (obviously it's my head, not the scope). How can I be sure to align it? Any tips?
Find a way to get in-person help from an experienced observer. +1 for Andrew's answers, but even if you get up and running with help here on SE, you'll learn tons by having someone actually standing with you at the scope at night. Best place to find knowledgeable observers is always your local astronomy club.
When you slew around, can you see anything at all? Random background stars? Let's hope so. If you see nothing at all, time to review things like removing the lens cap- or your scope was damaged in shipping, or some n008 fiddled with the adjustments that are not intended to be fiddled with (No offense. ;) ).
Can you slew the scope by hand and find the Moon? Look through the scope with one eye and at the Moon with the other. When you get close, you should even be able to just follow the sky brightness gradient as seen in the scope.
Select and center some recognizable feature of the Moon in the scope.
Is the feature also centered in the finder? If not, adjust the alignment of the finder. This is usually accomplished with a set of screws that hold the finder in place.
Is the mount level and is the polar axis pointed as close to Polaris as manageable?
It looks like this particular model [EDIT: DOES NOT FEATURE] a robotic go-to mount [EDIT: BUT SINCE SOMEONE WITH A SIMILAR PROBLEM MIGHT HAVE ONE...] Did you follow the alignment software procedure correctly? I don't know much about particular models, but the go-to scopes I've used ask you to center on a series of stars. You might need someone familiar with the sky to make sure you're getting the right ones, although the software should ask you to point at the brightest star in any general region, so a planisphere might be enough for you.
Andrew's covered most of your questions well, so I'll concentrate on the rest.
To test the alignment of you finder use a distant object during the day. A phone pole 50-100 feet away will work well, as will a well defined peak of a hill/mountain. Make sure it's NOT in the same part of the sky as the sun.
Point the telescope in the general direction of your target; but aim a bit low so you can follow it up to the top instead of being stuck looking at the sky.
Sweep your scope back and forth until you spot it in your finder scope, and center it roughly.
With your widest field of view eyepiece (20mm if you just have the ones that were sold with the scope) repeat the process in step 2 until you have your target centered in the eyepiece.
Swap in a higher powered eyepiece and repeat the process until you have the target centered as close to perfect in the eyepiece as possible.
Use the adjusting screws on your finder to center the cross hairs/dot on the target.
You'll need to check the alignment every time you use the scope; but unless you have to dismantle it for transport/storage it shouldn't drift much between uses. If you have to reassemble it, you'll need to repeat the process from scratch.
If you can't align the finderscope during the day because you're traveling to a dark site and don't arrive while it's still light things are harder but the same process applies. Generally you'll have to use a distinctive terrain feature on the horizon; either taking advantage of light pollution to make it slightly brighter than the night sky, in a completely dark location the horizon where it intersects with the milky way is probably your best bet since the contrast vs the ground should be clearest. The moon's an option if it's up (glare coming from one side in the eyepiece can help you find it once the finder gets you close); but beyond that I'm not sure what to suggest.
To balance your scope you need adjust it so the same amount of weight is on each side of the brackets holding it to the tripod. Do this with an eyepiece inserted (the weight will affect the balance).
Start by adjusting your scope with both axes locked so the tube is level with an eyepiece inserted. and then unlock the axis the tube rotates around and see what happens.
If the front of your scope drops down loosen the holding bracket and slide it back slightly, if the back drops slide the tube forward.
Repeat until it's stable when undisturbed, and when gently nudged drops just as fast in either direction.
Once you've done this, mark the placement of the tube so you can quickly align it again after reassembly.
Lock the first axis and repeat the process adjusting the placement of the counterweight.
"I have a Celestron Astromaster 114 EQ, which I saw was highly recommended as a great beginner scope."
Sorry, but unfortunately this is one of the worst telescopes a beginner could choose. The three major reasons are its flawed Bird-Jones optical design, which is incapable of high magnification and extremely difficult to collimate (align optically) which should be a routine maintenance procedure, its fussy, hard to use, and shaky equatorial mount, and its small aperture. The same money would have bought you a larger aperture Newtonian reflector on a simple solid Dobsonian mount. I'd seriously recommend returning it in exchange for a 6-inch Dobsonian if at all possible.
While some of the suggestions you've received are helpful, they would have been more so if people had actually taken a look at the telescope in question:
This is not a computerized goto mount, but a simple manual equatorial mount. Your telescope should not be swinging freely except when initially balancing it: there are clamps on both axes to hold it in place. Balance each axis separately: first the declination axis (by moving the telescope longitudinally in its cradle) and secondly the polar axis (by moving the counterweight up and down). Done systematically in the right order, these adjustments should take only a few seconds.
This telescope has some sort of weird "heads-up" finder, which should be adjustable. Personally I hate these things because they are so critical of where you place your eye; give me an old-fashioned optical finder any day. The essential thing in adjusting any finder is that (in daylight) you first point the telescope at something at least 1/4 mile away, and secondly, without moving the telescope, adjust the finder screws so that the finder points at the same object.
A word about lens caps. Many of these "beginner's" telescopes come supplied with a two-part lens cap, rather like a doughnut and its hole. Beginners often remove only the "hole" and leave the "doughnut" in place, blocking most or all of the light coming into the telescope. So, be sure you have removed both parts of the lens cap, so that the entire front of the telescope tube is wide open, at least 5 inches in diameter. In fact, glue the two parts of the cap together, since there is no reason you would ever want to separate them. [The reason for their existence goes way back into the history of telescopes, and has absolutely no relevance to anyone today.]
Well, I'm not sure if you are going to ever read this but, I have a solution and it will only cost about $40. I have the exact same telescope and like you, was having difficulty finding any object other than the moon. However, I received a wonderful Christmas gift this year and since it has made the telescope amazingly easy to use!!
It's a "Telrad" star finder. I mounted the base to the right side of the scope'sbrackets. It's made use of the telescope 1000% easier! I look to the sky with the cocentric rings, place the star or object/s in the second circle-right side, and BOOM! Centered in the eye piece. The hard part is focusing the object! This really isn't all that difficult as just a few turns of the focus wheel brings the object into clear view.
I've been looking at Jupiter this week. My only complaint now is the actual power. I'm debating getting better eye pieces but to be honest, I think the scope is just not "strong" enough. Either way, I'm enjoying my scope and use it nightly now. The "Telrad" has really saved my hobby.
I'm NOT paid by Telrad by the way, just an excited hobbiest! Enjoy.