Contrary to Karl's post, very few airlines (and certainly none of the budget ones) offer child fares. Children, from the age of 2, pay the full adult fare. Consequently they're allocated their own seats, together with the adult baggage allowance.
Because aircraft seat belts are only waist restraints (rather than the waist and chest restraints used in cars), they're just as suitable for small children as they are for adults.
Airlines normally carry pushchairs free of charge (in addition to normal baggage allowances).
Chris