You can survive DayZ more easily if you keep the following survival hints in mind.
Give yourself time to get the hang of things. Try for a server with a low population, to avoid survivors. Also, you want a server with daylight, if possible (you can also find servers that vary according to likely conditions in a given region). Playing on European servers at night contributes to likely daylight conditions, or playing on U.S. servers during local daylight hours has the same effect.
Go for supplies first. You need food and water, and particularly the latter. Find it first and you can avoid dehydration, which is a threat if you have to move long distances in search of a big city, for instance.
Focus on large cities. Small towns aren't worth your trouble and could leave you dehydrated, so check the map and find the bigger cities. Of course, bigger cities also mean more zombies, but it's better to run that risk while earning great rewards than it is to lose everything scavenging some tiny hamlet that yields nothing of interest.
Flee the zombies. Although zombies can run as fast as you can, they often won't chase you for long. You can also lose them by running around a corner and hiding in a shed or building, or just dropping to the ground (which also works great as an offensive measure of sorts, if you're carrying an ax and want to launch a surprise attack on a pursuing zombie).
Kill with guns. Guns are by far the most effective weapon. You can shoot a zombie in the head to kill it quickly. With a melee weapon, as noted above, you're best off if you lie prone. That way, you'll take less damage. Finally, do keep in mind that guns make noise and they could attract unwanted attention from additional zombies, so take that all into account.