I have a toddler and a baby on the way (in a few months). Can any other Filipino parents tell me at what age my wife and I can start traveling internationally with our kids?

Your age or the children's? LOL. But I would suggest after your child turned 1 year old so that all the needed vaccines have been completed already. We aren't sure what's really going on in that specific country you are going to unless you really go into detailed research prior to the trip and all we want is our children's safety and also after their protection all the time. Also, by that age his/her immune system is more dependable compared to the earlier months. ANd for your age, no limit at all! As long as you plan ahead, you have fun and enjoy the places with the kids and you make memories together :)