My daughter is 17 months old and we REALLY struggled with this for awhile. After lots of research and conversations with our doctor, these are some things to keep in mind when putting together your little one's diet: bananas, apples (and apple sauce), grapes, starchy foods(potatoes and even sweet potatoes), protein, sugary foods and dairy can ALL be constipating foods. You don't need to drop these foods all together but pay close attention to how much your kiddo gets. Mine used to eat a whole banana a day. Yikes!
Yogurt has probiotics/active live cultures that HELP digestion so DON'T cut that out. DO make sure that the yogurt you're giving your little one isn't full of added sugar or sweeteners. Apple juice is better than prune juice (and I've forgotten why the juice is different than the fruit itself or applesauce) and lots of 'p' foods help - peas, pears, peaches, plums, prunes. Also look for high fiber foods - broccoli is a big winner here, along with fortified oatmeal and beans beans beans (black beans, lima beans, kidney beans, chick peas, navy beans). We also added in water (instead of milk or juice) with snacks, and blueberries (which are affordable frozen, just don't buy anything with sugar added). Raisins,surprisingly, are also a huge hit around here. My kiddo thinks they're candy and will beg for more.
I didn't want to put additives of any kind into my kid's diet just yet, nor use softener or laxatives for fear that she would become dependent on these. Just being aware of the foods above and changing her diet to reflect this has made a WORLD of difference for us (just over the past 1.5 weeks!).
Good luck to you too :)