No Track Days as such, but your best bet would be to go somewhere like 3 Sisters Race Track, or even better (As its closer) Hooton Park by Vauxhall Ellesmere Port, and do a corporate event, or arrive and drive.
The karts they use are Twin Engined ProKarts, that push out about 11 hp...5.5 more than a standard single engined kart. That way, youll experience an actual racing track, with higher speed acceleration and the handling.
When it comes to the racing karts themselves, these have about 30 - 40 hp (Depending on the engine)..and the performance levels are ridiculous. I havent experienced it yet, as Ive not got the engine....but soon I will, cant wait

Karting as a hobby is good. Once youve got the equipment, £20 for a whole days track use becomes very desirable, as opposed to say £20 - £30 for 30 minutes. The racing itself is tricky, as you need a full licence (test can be done at Hooton Park)..and the kart and engine need to conform with strict rules set by the MSA.
A years worth of racing karts can be affordable, but prices can soon rocket, and on many an occassion, prices can zoom past those needed to compete in a season of Formula Ford !!!
Also depends on age and weight...