The earliest one you can get is the cleaver on the 2nd page, simply use that or any 6 dmg weapon you like and bash that attack button. If you're having a hard time, spam roll or just spam your ranged, which is ideally a shotgun or another ranged weapon that deals good damage. There is no balance in this game, some guns are simply better than others.

As far as i can tell melee weapons do not have different attack speeds, range or animations, so use any of the 6 damage ones, the easiest one you can get being the cleaver, the best ranged is the rocket launcher dealing 10 damage for 2 ammo, until you unlock that you can use the shotgun which deals 9 damage for 3 ammo.