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How to choose a telescope sensibly.  If you can read and have a modicum of common sense, read on!

Binoculars e.g. 7 x 50 or 10x50 are great for ‘browsing’ or finding brighter objects. They will also help in pointing a telescope in the right direction. So do consider these, as well as buying a telescope.
They are also VERY good for wide starfields that don't fit into a telescopes field of view. The other help binoculars can give is learning to navigate the night sky, and finding things you might want to view with a telescope. Larger binoculars eg 15 x 70 or bigger are excellent but these can be very expensive and do need a tripod.

Nearly all telescopes have finderscopes - small viewers used to accurately point in the direction wanted/ needed. The field of view, i.e. how much sky (land) can be seen is very small in nearly all telescopes.  You really can't point a telescope accurately i.e. find things, with a poor finderscope.
Finderscopes do need to be a reasonable size.

Most telescopes are intended for astronomy use, and give an image that is reversed vertically (upside down) and some are mirror imaged as well.

Some cheap telescopes claim to have an erector eyepiece for land/ sea viewing. These are invariably USELESS, as are some very cheap telescopes in general.
More expensive erector prisms used in better telescopes are fine for terrestrial (land) viewing.

The MOST important factor in a telescope is the SIZE of the large object LENS (or TUBE opening with telescopes with eyepieces near the top at one side.)

This determines the amount of light gathered, how bright and clear the image is and how much magnification can be used.

A   6” or 150mm  telescope will beat an 80mm or less telescope EVERY time.
You will also be able to observe much fainter stars and deep sky objects!

Ordinary telescopes of less than 60mm are virtually useless for any serious kind of astronomy use. FORGET claims of e.g. 525x magnification unless the telescope has 12” wide lenses or mirrors.

Most astronomy is done at 30x - 40x magnification or thereabouts. The field of view/ image brightness in smaller telescopes is ridiculously small at any higher magnification, as is the image brightness.
You MIGHT manage 15x - 20x magnification in a very cheap telescope!

The SECOND next important factor is a STEADY support, or almost any magnification is completely irrelevant!
The slightest attempt at focusing on a poor support, will shake the scope for 10 seconds or more, as will the slightest breeze!

The THIRD important factor is the quality of the eyepieces. Medium priced eyepieces can make a good improvement in all but the cheapest telescopes.

They will usually be 0.965” or 1¼“ fit.  1¼” eyepieces give more light transmission. Telescopes with 2” eyepieces are usually prohibitively expensive.
  You will only need two or three eyepieces e.g. 30mm, 20mm and 10mm for most viewing conditions.

A GOOD 2x or 3x Barlow lens will increase magnification very usefully on better telescopes.

Good telescopes give stars, even brighter stars, as pinpoints of light and have far less false colour or colour fringing. They also take higher magnifications far better.
This is easy to check with brighter stars.

The FOURTH factor is SIZE and WEIGHT of the telescope. You need space to swivel/ rotate a telescope and if you live in a light-polluted area, you may wish to transport the telescope to a darker sight. Smaller and lighter are easier to transport, up to a point.

The FIFTH factor is ease to set up. An equatorial mount needs pointing the polar axis due north (e.g. compass, pole star sighting) and initial latitude adjustment and levelling.  Dobson type mounts (lateral/ vertical mounts) don’t need initial adjustment which is an advantage for beginners.
Motor drives make viewing easier and more comfortable, but need a power source which may be heavy.

Equatorial mounts only need rotating the telescope once an object is in view, but do need careful balancing. With vertical/ horizontal mounts to track an object in the night sky the telescope has to be moved in two directions to compensate for the 'circular' motion of stars etc.

Computer tracking mounts often need a computer laptop attached and in any case, sighting of at least two, sometimes three recognised bright stars before they can be used, each time they are set up. Unless they are very accurately made and configured, the view may be near (but not actually in the eyepiece) for selected objects, which can limit their use.

Some EXPENSIVE computerised mounts are superb, but mostly, you can get a £150 telescope with a £150 GOTO mount, or a far better £300 telescope with maybe twice the aperture, that you point manually at objects to view.

The type of telescope (Refractor or straight through viewing, Reflector uses a large mirror, or compound (lenses and mirrors)) is far less important than APERTURE SIZE and QUALITY of the OPTICS.

Refractors are viewed straight through and need little adjustment. Reflectors reflect the light via a large mirror to an upper eyepiece but need periodic adjustment and possibly mirror re-silvering every 10 years or so. Cassegrain types use a mixture of lenses and mirrors to produce a more compact instrument. Refractors tend to be heavier and larger for a given size. Reflectors are often the cheapest option for a given size and light grasp. Cassegrain types tend to be expensive and have extra light loss..

Finally, the F factor eg F5 or F8 or longer etc decides the field of view size. F5 are called wide field scopes and allow, for the same magnification, more sky view. Basically the longer the telescope the greater its F value.
Do remember that whatever the F value, for a given aperture size, the same amount of light is collected!

Put simply, a longer focal length eg F12 and hence smaller aperture ratio gives more magnification for a given eyepiece and slightly less bright image. But this isn’t a disadvantage mostly, use a larger ratio eyepiece for general viewing!  

The main disadvantage  of smaller focal lengths and larger F values is that they show up poor optics far more than longer telescopes. In cheaper ‘wider’ field scopes, colour fringing, and various distortions to the image make viewing far less satisfying. 
Short telescopes with poor optics leave very limited options for magnification as smaller eyepieces have a very restricted optical throughput.

The other problem with cheap scopes is that the optical alignment can be so poor that they can be difficult to view through even at all!

Cheap telescopes have large internal opening stops (like large washers) inside the tubes which restrict light to the better central portion of the lens, but also severely restrict the light grasp to much smaller than the apparent size. A cheap 70mm refractor might have an actual usable 50mm aperture.

How to work out Magnification
Magnification equals focal length e.g. 800mm divided by eyepiece ratio e.g. 20mm
eyepiece would give 800/20 = 40x magnifying.  A 400mm scope with this eyepiece would only have 400/ 20 = 20x magnification.

A neat option on larger scopes is to use a limiting dustcap with restricted aperture(s) for viewing bright objects eg the Moon, Venus, Jupiter and Saturn. this also uses the best part of the lenses or mirrors.

Refractors should ideally be at least 70mm - 80mm ie 2 ½”- 3” minimum size. Reflectors are best 150mm  6” or above, though there are some very reasonable and optically sound 113mm 4” or 4 ½” models around.

AVOID refractors of  50mm 2” or less and reflectors of 76mm diameter. They simply don’t have enough light grasp for astronomy use.

There are some good fairly reasonable short tube 60mm or 70mm for star field or bright nebula viewing, but these won’t take higher magnifications for viewing planets etc. Again, the better ones are significantly more expensive say £40 +

If you wish to try to see deep sky objects such as galaxies, then a 6” reflector or 3 ½”- 4” refractor and a darker viewing site are needed. Even then they will mostly appear as fuzzy blobs! But finding and recognising such are a large part of the pleasure of this kind of astronomy.

To put matters in perspective, even with expensive amateur telescopes, you won’t see colour in nebula without enormous (15”- 20”) reflectors, you can see a wealth of detail on the moon with a reasonable 80mm refractor, and a 6” reflector will show a large number of much fainter objects and give reasonable (if small!!!!) views of the planets showing some detail such as divisions in Saturn’s rings, belts on Jupiter and Mars’ polar caps.

A modest 4” reflector will show Saturn’s rings and some detail on Jupiter.

70mm lenses from 15x magnification up, can show the phases of Venus, and Jupiter’s four moons.
Planets will show as disks but without much detail.

Viewing the moon and planets from your back garden in town /city skies, a good refractor of 70mm + or a reflector of 110mm + probably won’t disappoint. For more serious astronomy, you may want something a little larger.

If you want to continue with astronomy, do buy the best telescope you can afford /be able to set up/ transport reasonably easily.

Incidentally, MOST Chinese telescopes come with excellent, if not top top quality optics. You can get a comparable telescope for a quarter or a fifth the price of e.g. US scopes.

The best optical quality telescopes are VERY expensive, typically many hundreds to thousands of pounds.

A reasonably decent 6" reflector or 3" refractor shouldn't set you back more than about £140, but if you can afford more, you really need to spend AT LEAST £300- £400+ for a better telescope. It's also far better to upgrade APERTURE and upgrade eyepieces.

A very good option if you can afford it is to have a smaller eg 80mm widefield scope for use on the many nights when there is haze or turbulence and a better larger telescope for the occasional nights when viewing conditions are very good!

Remember that 8" and above reflectors, and 6"  +  refractors are sizeable and heavy telescopes.

You should also budget for sky charts, books, and a red light torch!