Golf battery polarity

Fitting batteries to any vehicle large or small you use the same basic rules.
You ALWAYS wire batteries positive to negative, (+ to -, or red to black).
You NEVER connect more than one wire to each battery terminal post.
The batteries on a golf buggy run in a basic 'serial' setup. You link the batteries together as a chain. The batteries and cables should 'snake' across the battery box with an open positive terminal post at one end and an open negative terminal post at the other end ready to connect to the correct leads on the buggy.

all of this work when all you need is the correct LED units
bulbs work which ever way the polarity is and bulb holders (sockets are always positive to the enter with the ground being on the outside)
by reversing the polarity at the battery terminals , other items that require a positive feed only , like the ignition module , converter etc will burn out
Flashers work through a relay and by reversing the polarity the coil used to close the points will not work
Replace all of the units that you have damaged by reversing the polarity and just connect the led 's up as you would a normal bulb
All of the rest is from you very limited knowledge of electrics

Even though the charger is new, I would check the charger with a meter to check its polarity just in case.

a lead acid will not charge with the polarity reversed
What happens is that in dead shorts and can destroy the battery
it can blow up the battery , blow the charger fuses and destroy the charger
Depending on the damage to the battery cell plates it may be recoverable

If the battery is not maintenance free, check the water levels in each cell. If you need to add water be sure to use distilled. Never use tap water unless an absolute necessity to get the vehicle running as it will destroy the battery.
Make sure the charger is set to 12v. Some chargers will have a 6v setting that typically are used in golf carts, boats, off grid solar.
Check the terminals on the clamps for corrosion.
Be certain that the clamps are tight on the battery posts.
If everything checks out, get a new charger.

check for clean shine terminals and battery posts . Check ING fuses and fusible link . Have you put the battery in with the correct polarity to earth. Have you checked the battery ( replacement ) with a load tester to see if it has a dead cell as well.

I suppose this is what you want to know
The battery charger has 2 leads coming off it as well as the power lead to the external power source .THE TWO LEADS WITH CLAMPS ON GO TO THE BATTERY THE RED GOES TO THE POSITIVE TERMINAL PLUS SIGN AND THE BLACK GOES TO THE NEGATIVE TERMINAL.MINUS SIGN, THE POWER LEAD PLUGS INTO THE POWER. AND THAT IS ALL THERE IS TO IT..

I presume the two leads you mention are:-
a) very heavy main +12V power lead from the battery terminal with approx 3/8" eyelug terminal,
b) less heavy wire that connects battery power to main vehicle fusebox for power distrubution- this cable may incorporate a fusible link.
The two leads (as above) connect onto the same heavy duty 3/8" terminal on the starter motor.
The lower 3/8" power post (closest to the motor housing) connects only to the starter motor and should not normally be tampered with.
There should be one more light duty wire that operates the starter motor engage solenoid .- This is connected back to the ignition switch 'Start' +12V control output.

You need a 12v lead to the ignition switch, a 12v lead from switch "run" terminal, to coil battery terminal. You also need a lead from the ign switch start terminal to the small post on the starter solenoid. Battery connects to engine block (-) and large terminal on starter.(+) This will give you start and run. Good luck...Hope this helps you!

Inspection: Connect the motor terminals directly to battery voltage (12V) and check that the motor operates smoothly. Next, reverse the polarity and check that the motor operates smoothly in the reverse direction. If the operation is abnormal, replace the motor.