Infants travel for free (on lap) on domestic flights.

For international flights, you have to pay a small amount + taxes, or points (was 5000 the last time we did it a couple years ago).

If you want a seat, you have to pay for it. In some fares you might pay 75% of the adult fare (child fares).

Regarding baggage, you have to check the airline policies for those routes, as they change. Allowance is usually similar to that of an adult. Check also whether strollers and car seats are counted or not.

The useful thing to do with the stroller is to "gate-check" it. In that case you leave it at the door of the aircraft and they give it back to you when you deplane, not counting towards the baggage allowance.