Hi Ashleigh - welcome to the forum!
It is always very wise to check out a complex system like ARC Pro well before the big day. Yes, the controllers require 2 x AA batteries in each - I recommend good quality alkalines, like Energizer or Duracell.
The Sunset Speedway set comes with everything you need for racing three cars. If you want to race four cars, you will need to add a fourth digital car (for example, a Digital Plug Ready car + a C8515 digital plug) as well as a fourth C8438 ARC Air/Pro Wireless hand controller. It is true that a second P9300 transformer is recommended to run more than three cars. Running four cars with just one transformer may be rather marginal on power. For five or six cars you will definitely need a second transformer.
I suggest getting the layout going without the app to begin with. Use the ARC Pro Quick Start Guide as your reference point. It explains how to connect cars, controllers, how to place the pit lane and powerbase, use the lane changers and pit entry etc. There is also a small switch on the left side of the powerbase which toggles between the standard digital mode and the optional 'analogue' mode (traditional one-car-per-lane running for cars without digital chips). In analogue mode, the lane changers will not function. I wonder if you have your powerbase in analogue mode? Make sure the switch is to the right (digital). Analogue mode only works with the red and green controllers, which control the lanes rather than the cars
Only when you have the hang of the basic powerbase settings and procedures in the Quick Start Guide should you move on to the app. Information about the app is quite basic in the Quick Start Guide, but is a good basis to start exploring the app. If you need more, I have written a guide to using the app, which is hosted on the SlotRacer Online website. The guide focuses on learning the functions of the app through hands-on racing. It might be worth reading: https://slotracer.online/missing-manual/
Finally, although it is possible to link a 1990s Scalextric set to a modern Scalextric Digital layout (via the C8222 converters), I wouldn't recommend it. Scalextric Digital requires very good track connections, which the current Sport track provides (with good maintenance). The older (pre-2001) 'Classic' track is more temperamental, as well as having a shallower slot depth that the latest cars have problems with. To build a bigger Scalextric Digital layout, I would stick with the Sport track.
I hope that is helpful.