A few ways to go though we are not swimming in choices.
NJMP: has rental leagues. A lot of fun but karts are old.
https://njmp.com/rental-kart-league/
Ovrp: A track/club just over the border in NY. No rental leagues but they do have rental karts. A good place to start owner-racing as they have a club spec margay lo206 series. Cheapest way to get started in owner karts.
Englishtown: Home of the F-series club and there are rentals available from full tilt racing on race weekends (Saturday). Call to reserve. You also have to buy a pit pass when renting at etown. In my opinion, not really a rental scene as the karts are only there when Keith has butts that reserved them.
K1 speed or RPM or whatever it’s called these days in Jersey city. Electric indoor. Had plans for a series then covid hit. Call them maybe.
There are a number of rental places indoors across NJ. Some are better than others. Also worth a mention is the crazy NY indoor place in Mt Kisco, GPNY.
UPDATE: Supercharged Edison has a time trial league that runs 2 races a week for 6 weeks. You go either tues or weds to set your laptimes. I believe that this will be speed 4, unlike their regular rental sessions which operate speed 3. Think it’s about 500, not sure. Just registered and will do a full review. Email them as it wasn’t showing on site but they responded that yes, there’s a league.
EDIT 2024: Since this thread comes up in google searches for Supercharged, adding links to Supercharged League reports from 2023.
ALSO:
K1 Speed now has a very interesting rental race series that has a national championship. Race the series to become champ of your facility to get to finals on the new big track out in California, I think. A fella I race a lot, Alex Smith, races this series and says its a great series. Google for NJ locations. One in Cinnaminson.