You're getting some great advice here.
Just to clarify what's involved with traveling with children, there are some factors to consider. The idea is that this might help you long-term plan and see more clearly where you want to head now.
I often recommend doing a trip with a baby around 4-5 months old. It'll never be easier. They rarely crawl that young (one of mine did at 5 months) and they don't need baby food. They're not exploring their surroundings yet and are content to just stay with you. But at the same time, you probably don't want to go somewhere they medical care is not up to standard. Little babies can get sick and you may not want to give them endless vaccinations at that age. Don't discount developing countries! Often, their elite classes and expat communities benefit from state-of-the-art facilities, in places you wouldn't normally expect them.
Once a child hits 4-5 years old, they become a lot easier to travel with. They are old enough to absorb their experiences and can ask questions and learn from visiting new places. Their memories will be a bit "uneven" ("Remember that cool artwork on the sidewalk...??") but it sticks.
Another factor, which I think is important is how many and how far apart your children will be. I was an "older mom" so I couldn't sit around and space my kids apart more ideally. They are just over two years apart and with three of them, that is very limiting, as far as hotel rooms and physically getting them around when they were little. I did it, especially short trips but while possible, maybe not ideal. If we had stuck to two, we would have been a lot more mobile. People who have a big gap, four or more years, have a real advantage. The older one can visit the restroom, read a book and play computer games while the physical care is more concentrated on the younger one. By contrast, I was trying to potty train the oldest, while breastfeeding the baby and keeping the toddler out of the cupboards all at the same time. During that period, we stuck to AI's!
Some destinations could be visited with small children but they might have factors that still would put you off for reasons you can't obviously see. For example, car travel in a country that has dangerous road conditions, without any good public transport options. Or going somewhere that is politically unstable. Friends got stuck in Guatemala once because of rioting, not as tourists but when they were adopting their second child. They were soooo glad that they had left their older child back in the U.S. with family! Even if you're not in actual danger, an experience like that could be an adventure for an adult but a downright scary one for a young child. Also, the medical facilities cited above and what has been mentioned, crime.
When you hear about crime and terrorism problems, it's worth looking into it. Where is it occurring? Who is the target? Places like Israel will quickly close anywhere dodgy for tourists, who are rarely the targets of any terrorism. Places local Israelis frequent are more likely to be hit. Mexico is another good example. For a long time, the crime stayed in certain areas easily avoided by visitors. When the violence spread and that was no longer the case, people stopped going.
When you have children, you become very protective of them and judging what is and what is not a "threat" can be a really difficult subject to evaluate. You will meet really overprotective parents as well as really "cool" ones and what works for one, is completely unacceptable for another family.
The last criteria is less a factor but cost. It will be more expensive but don't make this the deal breaker. If the flight is really long, or the best season is very short, you might want to pick that place. Unless you plan to homeschool, you'll find that later on, even when your children can travel well, you'll be locked into school breaks, which can be amongst the most expensive times to go anywhere. This kills me! All those great travel bargains and all the great places to see in the fall!!! Where do you want to go, ideally visited during term time?
