The magnitude or visibility of stars is measured by a system which means that the lower the number the brighter it is. A bright star like Vega of magnitude 0 is 2.5 times brighter than a star of magnitude 1, which is itself 2.5 times brighter than a star of magnitude 2, and so on.
Under really dark skies if you have good eyesight you should be able to see stars down to about magnitude 6, which is 100 times fainter than a star of magnitude 0. But in most towns and cities in the UK magnitudes of around 4.5 - 5.0 will be more the norm, of you are lucky. Judging the magnitude of stars can be difficult for beginners. A useful tip, which I found in the Stargazers handbook, is to use the four stars which form the pan of Ursa Minor, which is the constellation containing Polaris. These four stars have magnitudes of about 2, 3, 4 and 5 respectively.
If you suffer light pollution to any extent it may not be easy to find your way around the night sky. But some constellations are readily recognisable and it is well worth learning the names of their stars. Thus in the winter you have the Orion Nebula with the easily recognisable reddish Betelgeuse at the top left, Bellatrix of Harry Potter fame over on the right, then the 3 stars of the belt, Alnitak, Alnilam and Mintaka, then the brilliant blue-white Rigel bottom right with Saiph making up the other corner.
Not far above Orion is the brilliant yellowish Capella in Auriga and to the left you have the twins Castor and Pollux and not far below Procyon whilst the brightest star in the sky, Sirius, is just a little further down and above Orion is another reddish star, Aldebaran. In the Autumn the square of Pegasus,, consisting of Alpheratz, Algenib, Sheat and Markab form another easily recognisable asterism and is the starting point to finding the Andromead Nebula. And in the summer you have the summer triangle of Vega, Deneb and Altair.
By being able to identify these stars and constellations you may then be able to use a technique known as Star hopping. This is a method by which you first find a bright star, then either move a set number of degrees and/or move to another star, and so on, until you reached the object you are trying to find - maybe another star or a more likely a galaxy or a nebula. This is not always easy for the beginner in light polluted skies, especially if you have a manual scope but it can be learnt if you practice. If you have a mobile phone there are a quite a few apps which can be downloaded and, by simply pointing up to the sky will tell you what is there. Another way is to use a planisphere.
Constellations: From ancient times man has looked at the skies and seen patterns. Those we use today are based on those of the Greeks, with minor additions during the last few centuries for the Northern Hemisphere and rather more for the Southern.
The visibility of the constellations will depend on your latitude and the time of year. Some constellations such as Ursa Major will always be above the horizon and these are known as Circumpolar constellations. Others such as Andromeda, containing the famous Andromeda Nebula, are winter constellations, and of course Orion, probably the most viewed constellation and one which all beginners will look at.
It takes a lot of practice both to get to know the constellations, judge distances and to get used to determining the magnitudes of stars.
Double Stars: Perhaps as many as 50- 75% of all star systems are multiples, systems where 2 or more stars are in orbit around each other. And then there are the stars which are not physically connected but appear close together as viewed from the earth. One of the most famous doubles is the naked eye one of Alcor and Mizar in Ursa Major - but you need a dark sky and good eyesight to see it. It is a good object to view in a small telescope and will give you confidence in finding objects. Another good double to look at for the colour contrast is the blue and gold of Albireo in the constellation of Cygnus.
Variable Stars: These stars brighten and fade over variable periods of time. Eclipsing Binaries, such as Algol, do this because one star appears in front of another as seen from the earth. Other are true variables whose light varies for a variety of reasons. One of the problems in the UK with observing variable stars is that you seldom get sufficient successive clear nights in order to properly compare and observe the changes in brightness.
The best way to decide what stars to look for is to use the monthly star guides from magazines like 'The Sky at Night' and 'Astronomy Now'. Many astronomical society web sites will also have similar monthly features.