All family members travelling abroad must have a valid passport. Apply for a child passport if your child is under 16. If they're 16 or over they need to apply as an adult. Allow up to three weeks to receive a new or renewed passport, though there are (more expensive) fast track services available.
Child passports are valid for 5 years, adult passports for 10.
Note that many countries require passports to be valid for at least six months from date of entry. Check the UK Government's foreign travel advice site for information about each country, including visa requirements.
If your passport is lost or stolen while abroad contact the local embassy or consulate, report the theft (if stolen) to the police. You can apply for an emergency travel document or 'emergency passport' to get home, allow two working days.
Visa requirements vary by country, cost and duration. Up to date information is available on the UK Government's foreign travel advice site. Select the country you are considering for your family holiday and then 'entry requirements'.