Look in the night sky to spot this comet in June & July
Comets have long had a great deal of fascination for humanity when they are visible in the night sky. On July 1 a comet with a long name (C/2013 X1 PanSTARRS) will reach maximum brightness and we will have a chance to see it with our naked eyes.
Photo of a comet courtesy of Soerfm @ Wikimedia Commons
Comets are balls of ice and maybe a little rock. While not that much is known about comets they generally are thought to originate in the outer reaches of the solar system in the Oort cloud. It is believed that they are knocked out of this cloud and are then pulled into a long eccentric orbit that goes close to the sun.
Image courtesy of Юкатан @ Wikimedia Commons
Comets are mostly ice and as they approach the sun they heat up and vent gas into space. This creates the characteristic "tail" that we associate with comets. However the tail doesn't follow the comet but instead points in the direction of the sun.
Ancient people's thought comets were warning signs that predicted some great disaster. Of course the only major disaster is going to be if the comet hit us, which this one is not going to do (no need to call Bruce Willis to blow it up).
Public Domain woodcut by Jiri Daschitzsky of The Great Comet circa 1577
On and around July 1, this comet will be bright enough that you can see it with the naked eye. However this will need to be on a moonless night away from city lights. On July 1 the moon will be waning and won't rise until 2:54 am. By the weekend it will not only be mostly dark, but will rise only a few hours before dawn. These are near perfect comet spotting conditions.
It is already in our skies and you should be able to spot it with a telescope or even strong binoculars. Even a fairly standard 10x50 mm pair of binoculars should be more than enough now to see the comet. As it approaches the easier it will be to spot.
During May the comet is in Aquarius, in June it will cross Piscis Austrinus (the southern Pisces) and in July it will be in the minor constellation Microscopium.
Image of Piscis Austrinus courtesy of Allthesky.com
Generally you will want to look for the comet early in the morning before first light. I recommend downloading a night sky app for your phone to keep track of where the constellations are at any particular time.
C/2013 X1 PanSTARRS is a doubled tailed comet, however from the angle that we will be viewing it; it won't particular look like your classic comet, but instead appear like a big ball of cotton wool.
Image explaining 2 tailed comets courtesy of NASA Ames Research Centre
One of the best ways to view a comet is to look up your local astronomy club. Most hold events every month, both in major cities and out in the countryside. They will usually have an array of telescopes setup to view an event like this.
Photographing a comet
You should be able to photograph this comet with a DLSR or mirrorless camera. For comets you want the longest lens you have (or can borrow from a friend). A 100 mm lens would be the minimum, but the longer the better. With a 100 mm lens you want an exposure no more than 6 seconds. With longer lenses the exposure length can be reduced.