You need a motor controller, specifically an h-bridge (for a brushed DC motor) capable of handling the current (60 amps) - you need an h-bridge so you can control the motor in both directions, as well as the speed.
An h-bridge of that size will not be inexpensive - to give you an idea:
https://www.dimensionengineering.com/products/syren50
...and I can't even recommend that because I don't know the full specs of the motor. You say it's max current draw is around 60 amps - this very well might be correct, but I don't know what its running current under load is - but likely, if everything you have said is correct, that motor controller would probably be perfect.
Another option would be to go on ebay and get one of the variable speed reversible 24 volt motor controllers for the wattage of your motor. It may or may not be cheaper, but ultimately, these controllers have a manual switch soldered on to change direction. They use a potentiometer to control speed, via what I believe to be a 555 timer. You could bypass (read as "hack") that to supply the PWM to the output mosfets via the Arduino. For the manual switch, you could get a high-current reversing relay/solenoid for an off-road winch, and control that with the Arduino as well. Caveat: Only switch the direction of the motor after at comes to a stop (otherwise, you will weld the contacts of the relay - in the best case). Of course, this option is probably beyond your current skill level right now.
You do not want to try to switch the current of this motor directly (ie - using a switch, or a relay configuration as an h-bridge) - without some kind of soft-start/stop capability, because the current level it draws is so high. Also, it is not a toy motor, so treat it like you would a go-kart engine or similar - don't wear loose closing or long hair around the spinning shaft, don't try to stop the shaft by hand, always mount (clamp) the motor securely before applying power, and when you are testing your creation (ie - playing with the Arduino code while the motor is under power) - do so with the wheels of the contraption off the ground, in case you have a "runaway" issue.
These motors can run on lower voltages, but won't have nearly the torque as it would at 24 volts. Remember, they were designed for a mobility scooter, and could likely move that scooter at 10-15 kph for many kms, before needing a recharge, all the while using SLA (lead acid) batteries and the weight of the person (plus the scooter). I can say all of this because I have more than a bit of experience playing with such chairs and scooters from helping a friend who refurbishes them for donation (and we build robots with them on the side).
Your batteries - no matter how you look at it (even if you used SLA) are going to cost more than a bit of money themselves. A LiPO pack (likely 28.8V) won't be cheap - plus you still need the charger. Personally, I would go with LiFePo drop-in replacements for SLA, unless size or shape was an issue - but they aren't cheap (they are very lightweight, tho). Whatever battery you choose, be sure to put a 50-60 amp fuse in-line with the positive pole of the pack, as close to the positive as possible (within 10 cm is ok). Such a pack, regardless of what battery chemistry you use, can easily become an impromptu welder - ask me how I know (and in the event that this occurs - that is, a direct short - with a LiPO battery, you better hope you have your insurance paid up, because fire and explosion will likely be the case - though that can happen with any battery, it is almost a surety with a shorted LiPO pack).
I know none of this is likely what you wanted to hear, but it's the truth. Large motors like these aren't cheap to control or run. That said, considering what you paid for it, you go a helluva steal - so consider that savings going toward the remainder of the project.
/note - I'm a yank, so my imperial -> metric conversions are -very- approximate...!