While light pollution certainly makes things more challenging, there are a number of objects that tend to compete with light pollution well-enough to be enjoyed.
I have done astronomy outreach events from urban parks (located in downtown metropolitan areas) where light pollution is extreme. Faint deep-sky objects typically can't be viewed without special equipment -- but there are objects that can still be viewed.
Specific planets such as:
These bright planets are usually easy to spot if they are in the sky. Venus is so bright that it can be viewed in the middle of the day (the trick is knowing precisely where to look). I have been able to show Venus to school students in the middle of the day. More impressive yet, is that once the students see it, I'm able to have them put their face near the side of the telescope -- to use the telescope as a guide. The students can see Venus as a point of light in the sky without using any optical aid at all (no binoculars or telescope). But if they turn away and look again, they usually have difficulty finding it again without something to point the way.
Jupiter is another object that can be seen in mid-day ... although not as bright as Venus and this one requires a telescope (you can't see it with just your eyes -- as is the case with Venus). It appears fairly washed-out with very weak contrast (as if looking at it through a fog) when observing it in mid-day.
If these planets are visible even during mid-day, they are certainly also visible at night in light polluted conditions.
I did not mention
Mercury is sometimes visible ... that object is tricky at times, but at other times it is easy to spot (it is not nearly as bright as Venus).
Uranus and Neptune are far enough away that don't appear brighter than most other stars in the sky (they don't stand out like the closer planets).
Deep Sky Objects
Experienced observers have an easier time finding these objects in light polluted skies mostly because they know what they are looking for ... by which I mean for each object, they have a pretty good idea of how it will appear in the eyepiece of the their telescope. Many deep-sky objects will appear as a faint smudge that an inexperienced observer might ignore -- not having realized that this was the object. Experienced observers also know where to look to find each object -- so they aren't searching randomly.
There are many "planetarium" applications made for smart-phones that tie into the phone's GPS, compass, and level/motion sensors so they know where you are and where you are pointing the phone in the sky. These make it MUCH easier to find objects.
Bright stars do tend to be visible through a telescope -- even in light polluted skies. This makes certain open star-clusters good targets.
The Pleiades cluster (located in Taurus and tends to be visible during months such as December, January, & February) is one of the brighter open clusters and is usually very easy to spot -- even in heavily light polluted skies. Many open-clusters are visible from light polluted skies. These objects also tend to cover a patch of sky large enough that they are often best observed at relatively low-magnification.
The Orion Nebula (Messier 42) is a particularly bright nebula. It is so bright, that it appears star-like even without using a telescope -- even from urban locations. Through a telescope it appears as a large gas cloud. The core of this nebula is particularly bright. In dark-skies, the object is so large that it can fill the eyepiece. Heavily light polluted skies tend to wash out the fainter and more extended areas of the nebula, but the bright core is still visible -- just smaller (and less impressive).
The Ring Nebula (Messier 67) is another nebula that tends to be visible even from light polluted skies. Through a telescope it appears as a tiny donut-shaped cloud (like a smoke-ring) -- but is bright enough that it often can be viewed. This is a fainter object ... but I usually am able to find this from urban skies.
Depending on where you live, there may be several southern-hemisphere objects (I am a northern-hemisphere observer).
The only galaxy I have managed to view from light polluted skies with ordinary equipment is the Andromeda Galaxy (Messier 31). The core of the galaxy competes rather well with light pollution ... but not the galaxy disk. This means it wont appear as a galaxy... it will appear as a fuzzy pin-point of light.
I have technically been able to observe other galaxies from urban locations -- but only with highly specialized equipment.
Here's an example...
This first image is a heavily processed image of the Andromeda Galaxy (Messier 42). You will never see an image that looks like this through any telescope. These are my own images.
Here is a mostly unprocessed image out of the camera to show roughly what you might see from a dark sky site. Notice you can see the disk ... although it does become quite faint even in a dark sky site.
Here is the same image again, but I've processed it in an effort to simulate light pollution. I think I probably did not add enough haze to this image for the simulation. When I observe from light polluted skies, I ONLY see a tiny pin-point of light surrounded by a haze and don't see even a hint of the disk structure (this image does show a hint of it). So imagine only seeing the pin-point in the center and that's what you'd be trying to find.
I usually do not see the tiny companion galaxies at all.
Nearly any telescope can also be used to view the Sun WITH A SAFE SOLAR FILTER attached. The solar filter looks a bit like a cap that fits over the front of the telescope. It blocks 99.996% or more (usually 99.999%) of the Sun's energy. These safe solar filters are usually not particularly expensive.
If you do this, be safe:
- Inspect the filter each time before placing on the telescope. Make sure the filter is not damaged and discontinue use with a filter if it is damaged.
- Remove or cover the finder-scope.
- Do NOT look through the telescope to point the scope at the Sun. Instead, observe the shadow cast on the ground and orient the optical tube so that the shadow of the telescope's optical tube is round. (There is such a thing as as solar finder...and some people make their own.)
- Be safe. Staring at the Sun without proper protection damages the retina of your eye ... but this works a bit like getting a sun-burn ... where you do not initially notice the damage until several hours later. Often the damage from directly observing the Sun without a safe filter may not show up until one or two days after the damage is done. This can result in the false-perception that you are ok ... only to find out you are having difficulty with eyesight a day or two later.
Do not use filters that attach elsewhere to the scope (such as the eyepiece) as these allow too much energy into the instrument and can cause damage.
If you do equip the telescope with a safe solar filter and practice safe viewing techniques, then you can easily observe sunspots.
This image was captured through a solar filter in October 2014 when one of the large Sun-spot complexes of the last solar cycle was active. The Sun appears orange in this image because the filter being used reduces blue wavelength light more aggressively than red wavelength light (if all wavelengths are reduced equally then the Sun will appear white).
Dark Adapted Eyes
Even in urban light-polluted skies, protect your eyes from light. If you can find a location where you are not able to directly see things such as street lights, etc. then your pupils will dilate and allow you to see fainter objects. Part of the problem is the light pollution -- to be sure. But the other part of the problem is not having dark-adapted eyes. Human eyes will quickly constrict when you view a bright light source... but take much longer to dilate open again. It requires something like 40+ minutes to become "fully" dark-adapted ... but you are more than 90% of the way there after just 20 minutes. Just a few seconds of glimpsing a bright light source can ruin that (and resets the clock back to another long wait to get dark-adapted again).
Some observers will put a sheet, towel, blanket, etc. over their head while observing at the eyepiece to protect their eyes from stray light sources such as nearby street lights, house-lights, etc.
Know where to Look
Use a planetarium app on your phone to help you located objects -- so you aren't searching randomly. On a mobile phone, Sky Safari is very popular (and cheap). On a computer, Stellarium is popular (and free ... but only on the computer. There is a mobile version but it is not free.)
While there are sophisticated "go to" telescopes that can quickly located objects once they are "aligned" to the celestial sphere (they need to know your location, date, time, and then you orient them by helping them center on a couple of bright stars) ... the electronics and motors increases the price of the scope. So depending on your budget ... they might not qualify as "low cost".
BUT ... there are ways to find objects by first locating something bright enough that you can find it ... and then determine how far and in which direction to move the scope. This would work with any telescope.
Telescopes are typically paired with a small "finder". Sometimes the finder is a tiny scope (very low power so it sees a wider piece of sky) and sometimes it's a red-dot finder (no magnification). But one interesting type of finder is the Telrad ... this finder has 3 concentring rings. The inner-most ring is a 1/2° circle on the sky. The next ring is 2°. The outer-most ring is 4°. The find is actually popular enough that most software and most star-charts support it. I have paper charts that come with a clear over-lay that has the Telrad pattern. Also a number of applications (such as Stellarium or Sky Safari) also support the Telrad overlay. The idea is that you use this with your chart/app to determine how many Telrad diameters to move the scope from an easy-to-find object to locate your hard-to-find target ... and then make the very same moves with the telescope using the physical Telrad finder.
There are numerous techniques and it takes a bit of practice, but in time you get fairly good at locating objects.