A healthy infant can typically start flying internationally after one week old, as long as baby’s pediatrician gives the ok.
On most domestic flights babies under two years of age fly free for infant-in-lap tickets, and most international flights babies under two years of age fly for 10% of your ticket cost plus any associated taxes. So, if your round trip flight is $1,500 you’ll pay $150 for baby to come along. Baby typically also gets one free checked bag on an international flights, in addition to a checked stroller/ car seat (but this varies across airlines), and gets one carry-on bag allowance (check your airline on specific infant in lap restrictions and baggage allowances).
Because of the way we travel, we often purchase one-way international tickets. Although most airlines still seem to charge 10% of the ticket price for an infant-in-lap ticket for babies under two, we have come across two airlines so far (Delta, South African Airways), that charge 10% of a round trip ticket for baby, so we ended up paying closer to 30% of our ticket cost. For some reason, these airlines have policies that do not ticket one way tickets for babies internationally, even if mom and dad are flying one way, and require the purchase of a round trip ticket. We avoid using those airlines for that reason, and it is wise to make a phone call before purchasing your ticket to find out the exact cost of what baby’s ticket will be.
I have had difficulty purchasing an infant-in-lap ticket online, particularly for international flights originating on U.S. carriers. Many non-U.S. carriers allow you to purchase an infant-in-lap ticket online, such as Etihad. Typically, I purchase my own ticket online, and once the ticket is confirmed I call the airline directly, request to purchase an infant-in-lap ticket, and provide them with credit card details over the phone. If I am traveling more than two weeks in the future, a paper ticket for baby is generally mailed to me, but when I have been traveling in less than two weeks a paper ticket has been issued at the counter for check-in (allow more time at check-in for this).
The best site I’ve found to search for cheap tickets is www.smartfares.com. I also like skyscanner.net as a great search hub for finding competitively-priced airline tickets on any airline under the sun in the world, and to and from any destination (sites like Orbitz require you to originate travel in the U.S.).
However, if you want a separate seat for baby, you’ll have to pay full fare for a seat. I have met parents who do not find it comfortable to fly with an infant in lap and appreciate the extra seat where you can put a car seat and where baby can sleep. If you can afford it, that’s an option, but an expensive one for frequent international travelers (and the only way baby will accumulate frequent flyer miles). We’ve had pretty good luck on international flights requesting a bulkhead seat with bassinet, which they reserve for families with young children, and provides an extra sleeping space for baby. We’ve also had good luck getting an empty seat next to us. Most airlines will allow you to bring a car seat on the plane for baby, as well, if the flight is not full and you are able to get an empty seat next to you for the car seat, which we have also done before. Additionally, once baby is two years old, your only option is to buy a full-priced ticket, plus by then it probably won’t be as practical or comfortable sharing a seat for 10 + hours with a wiggle worm.