Skipper, this is a tricky one, and you really ought to discuss it with your family doctor. The Center for Disease Control has an extensive (and complicated) article about it that may help you get a basic understanding;
Generally though, children over 12 months and under 2 years have had their first allotment of the basic immunisations, and that should be protection enough. The next lot of MMR etc isn't due until they're 4, but can be accelerated (read the CDC article).
As for Malaria and Dengue Fever, keep the kids clothed in long loose-fitting garments and use insect repellent on exposed skin, but be mindful of the EPA (USA) guidelines on insect repellent and infants ie keep it away from eyes and mouth) : http://www.ivillage.com/insect-repellent-safe-babies-and-toddlers/6-n-136820
Don't use tap water in any of those countries, make sure they drink only bottled water. Try to avoid getting too much water in their mouth when you bathe them. Take lots of anti-bacterial gel and hand wipes. about 7 years ago