Melee Weapons are good, but firearms are better.
After you have learned the principles of Gun Safety and possess a sound mind & cool temper, you may consider getting yourself a pistol as your first firearm against the living dead. They are easy to obtain legally and come in a delightful variety to suit your needs.
Made famous by cowboys and the Lone Ranger, they are renowned for their reliability, easy operation & maintenance. The only down side is that they carry limited ammunition and are slow to reload.
The minimum caliber you should consider is the .38 Special for its proven stopping power against human threats, undead or otherwise. I recommend the use of JHP or Jacketed Hollow Point ammunition. This type of bullet expands upon entering, allowing it to transfer more of its energy to the target by creating a larger wound cavity. As a rule, your ammunition should be able to penetrate 12 inches of flesh to reach a vital organ. For the undead, the brain is the only relevant organ. Smaller ammo will need more shots, but having a tiny pistol is better than none at all.
Shot placement, caliber, & skill will contribute to stopping power.
Higher caliber revolvers are also good, but you will not be able to conceal or bring them around easily. In pre-SHTF conditions, concealment will be very important because it will be intended to protect you against living & breathing (non zombie) bad guys.
Compared to revolvers, these carry more ammo, are much faster to reload, and have way more choices in model.
The 9mm caliber is one of the world's most popular ammunition types for personal defense. This is due to its accepted reputation for stopping power + higher capacity compared to larger ammo types. Ballistic tests and real-life experience has taught many a patrolman the comparable performance of the 9mm JHP to the much larger .45 caliber. In my opinion, the main benefit of the 9mm is that you can carry a lot more ammo in your magazines on a reasonably-sized pistol.
I use a Para-Ordnance .40 caliber pistol as my primary weapon. Its magazine can carry up to 17 rounds +1 in the chamber. I've got 50 rounds of JHP for both pistols and another 400 for practical shooting training. The revolver I'm using is an M206 from Armscor & serves as a backup weapon. If I had it my way, I would get a high-capacity 9mm for Mrs. Zombie Hunter so she can defend the home when I'm away.
Remember folks, practice and teach Gun Safety to your loved ones!