Karting: most go karting tracks operate some sort of free practice-type session, you just need to phone up and book it - probably around £20 for 15 minutes. To race you'll need to get a dozen or so friends together and book the track; depending on the format of the racing, it'll probably cost more than £40 each.
Cars: Most race circuits run driving sessions in a variety of cars - Elises, Ferraris, Formula Palmer Audis etc - which'll cost the far side of £150. I don't know of anywhere that allows you to actually race cars.
Whereabouts are you in the UK? It'll help to suggest some nearby tracks.