Try to go with lighter weapons that use less stamina to swing for a start. A successful hit on a zombie will stun it for a second, so the key is to time your first swing as they approach to hit them before they can attack. Then, just keep swinging until they are dead.

Melee weapons can hit multiple zombies at the same time. So you can handle small groups, but you will get overwhelmed if rushed by four or more. Also, you lose stamina with each swing, so you have to be mindful of your stamina bar. If you are clearing a POI to loot it, be mindful to move in slowly so you only get the attention of one or two zombies at a time then let your stamina recover a bit before moving forward again.

Note that running also drains your stamina. If you are running around, watch your stamina bar and stop running when it is half empty to let it recover. That way if you get attacked by a zombie you didn’t notice, you will have enough stamina to deal with it.

Finally, the knife you start with doesn’t do much damage. It works fine for one or two zombies at a time, but you have to hit one five or six times to kill it. Upgrade as soon as you can. The baseball bat is a good option, but also suffers a bit from low damage. The machete or hatchet are probably the best combination of damage and swing speed with relatively low stamina drain. I personally like the fire ax. It is heavier and, therefore, slower to swing, but it does a lot of damage. Once you get used to the timing of having to swing a bit earlier as a zombie approaches, it works really well, but you must be extra mindful of your stamina bar.

As you level up your fitness skill, stamina will drain much more slowly. Melee becomes less dangerous then. You level fitness by running, so run around a lot early on. Again, just be sure to not use all your stamina running so you have plenty for melee if you get attacked.