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Looks like a fun track!
Just a word about train transformers:. I used them for years and the cars run fine. No, it doesn't hurt the motors.
However....what you may find (as I did) is that their output is not filtered. It caused problems with the sensors for lap timing and scoring. When I put the filtered supply on the lap sensors, everything worked fine.

So: a filtered supply is still better.

The power supply that I posted a link to is $28.90 with free shipping. DC motors will not run on AC, they just make a humming noise and get hot. If you add a rectifier you will have DC of a sort and the cars will run on that. You can add a filter capacitor to the circuit. If you use a variable transformer you can vary the DC output, however the voltage will drop when a load is applied to the circuit. The best power supplies are regulated which eliminates the power surge that you get when there are several cars on the track and one of them falls off. Power supplies that come with sets are not regulated and the power surge caused by one car coming off can sometimes cause another car to come off as well. Besides using a single regulated power supply some people just have a separate set type power supply per lane, those are each good for less than an amp. If you need a lot of amps you will want a large, heavy and expensive transformer if the power supply is the traditional linear type.
There is a more modern type of power supply called a switching or switch mode power supply, those are more compact and less expensive because they only need a small transformer. A well designed switching power supply can have all of the advantages of a linear power supply without the cost and bulk.