First check what indoor tracks are available in your area, do a couple of races and see if you like it, and how well you do, then go back again some other day and see if your driving improved.
If you feel like the indoor karts are too slow for you, then it’s time to move on to the next level. Find a local track near you and ask if there is an arrive and drive/rental program, and start driving 4 cycle low hp karts just to get you started. This should take place at a real race track ( bigger than indoor)
Once you become better, and more addicted to the sport, and you can’t take it anymore, other than wanting to buy your own equipment, then it will be time to look around for a good used turn key and ready to go kart.
I wouldn’t be too concerned about adult classes, all full size karts are adult drivers, in most clubs you have to be at least 16 or 17 and up to 65 years old to race.
A great place to look for used karts, and parts is right here, there is always something available at a good price, and there are many racers willing to give you a hand and good advise. Good luck to you, and welcome to the sport. FastFreddy. ;)