Fines for motoring offences such as speeding should be increased from £60 to £90, the government suggested as it launched a consultation on the subject.
Transport Minister Mike Penning said current fines created a perception that the offences were “minor”.
In future careless driving could also become a fixed penalty offence with a “remedial training” option, he added.
Educational courses for other offences, such as not wearing a seatbelt, could also be offered.
The government believes that the current regime for dealing with careless driving - defined in law as driving that “falls below what is expected of a competent and careful driver” - is overly bureaucratic.
Ministers are seeking views from the public on the proposals, which would apply to England, Scotland and Wales, before deciding whether to pursue them.
Police catching careless drivers at present can issue them with a summons to court, but the government believes they are deterred from doing so by the “heavy burden of paperwork” involved.