FranticDan has already made some very good points, so I won't repeat those; but I will add some more that I use when I play:
Ladders in hotbar are a must (to quickly climb up walls) along with spike traps. Roof tops are great if you get into a pickle and crouching while using it to block their line of sight causes them to lose interest in tracking you down. Spike traps are great to get them to damage / kill themselves and slow them down. I have had times where my line of placed spike traps delayed some zombies that sensed me with Feral Sense on.
Inventory management is required - if you are encumbered, you are asking to become a zombie snack. Secure storage boxes ready to place down are a must. Only carry what you need right that moment, don't worry about building up resources for crafting later game items. If it isn't something you can use that minute, then it is better to come back for it later.
Crouch / walk most of the time. You need to spot the zombies first, getting that first sneak shot arrow hit is key - headshot is best.
Go slow at first, especially beginning game. Pick out easy POIs to clear out first. First week as I am trying to get armor, better tools, and weapons, I will only do one POI a day (60 minute days), unless there are two low level ones next to each other; but I always leave time to get back safely to my night location as quietly as possible and undetected - which may require backtracking - a lot.
If you are playing with the 4 easy perk points at the beginning, I tend to spec into Deep Cuts (Q2 bone knife and slow on bleed), Cardio (stamina regen when running), hidden strike (for extra damage), and parkour (for reduced jumping stamina). From the shadows is also a good one to get them to lose interest in your faster (along with archery for better bow and damage).
One zombie is easy to handle if you follow the above. Two is tricky, but doable. Three or more, it is better to use ladders you already placed in strategic areas to get out of harms way and try to get them to ignore you. Then pick them off with arrows (sneak damage). Zombie dogs - forget them. If you haven't seen me yet, back off unless you got a lot of arrows and a high ground they cannot reach.
Don't engage zombies just to get extra XP. If they are running away (like a wandering horde), let them be. Right then, they are not doing anything to you, so why not just let them go?

Don't get focused on that one zombie. Hit them and run away, keeping an eye for any additional Z's that get interested in what you are doing at that moment.
You make a mistake and it will cost you, maybe even to the point you die. But that is the whole point of Insane / Nightmare speed, to challenge you.