The best place we’ve found for booking cheap flights is Skyscanner. We’ve saved thousands by going through them vs. an individual airline. Detailed guide on airplane travel with kids coming soon. 🙂
Train, tram, and subway transit can vary a LOT by location, but generally speaking, you’ll want to get your ticket ahead of time (online or through an app if possible), and be at the station at least 30 mins early for long-haul or overnight trains. Some cities have day, week, or month passes that can save you a bundle, and kids under a certain age travel free (this varies by location as well, but we’ve found “under 6” to be fairly common). Be prepared to get on and off pretty quickly, and if you have little kids, make sure to hold their hands as you go over the gap between the platform and the train. Depending on the area you may or may not find stroller-friendly trains—but usually bigger cities are pretty good about having some. For instance, Vienna’s S-Bahn has handicap-friendly trains that come every so often (the rest are a couple steps up to get in), but all the U-bahn trains are handicap-friendly.
I personally hate traveling long-haul by bus, especially with kids, but local transit usually works out fine. We took buses a lot in Istanbul, and rarely had any issues. As with all public transport it can get really packed at certain hours, but if you plan your trips out correctly it’s usually not a big deal—and if you have a baby or little kids, people are usually really quick to help if needed. Peak times are relatively the same as you’d expect in a big city in the US—so first thing in the morning, lunchtime, and mid-afternoon to early evening. But again this will vary by city—you may not even notice the difference in less-populated areas.
Taxi is a more expensive way to get around, but may be preferable if you don’t want to have to deal with the potential crazy of public transportation. Uber is also in a lot cities around the world now. It’s not as common for taxi drivers to speak English, so you probably want to be prepared with either a translator or a map if necessary (or basic knowledge of the local language). And just a word of warning… Driving can be VERY different from what you expect in other parts of the world. 🙂 But taxi drivers usually know what they’re doing, and most of the time, you don’t need to be concerned.