Infants (0-12 Months)
Easy puzzles with big puzzle pieces are perfect for babes under a year old. My daughter mostly likes to take the pieces apart and suck on the knobs, but that's okay. Getting her familiar with puzzles is great to begin with. Also, we practice naming the animals and animal sounds when we play. You can even find simple puzzles like this at the Dollar Store.
The soft Activity Cube at the end is also great for babies. They are soft so they are safe for babies to throw around and suck on. There are different ones where babies can pull shapes in and out of the cube, introducing them to the idea of fitting objects into spaces.
Toddlers (1-3 Years)
Chunky puzzles are great for toddlers! They have large pieces and relatively simple shapes. You can find these puzzles in all sorts of themes to engage your children. Melissa and Doug also makes sound puzzles and my son LOVES these! He has a couple of the train ones that make train sounds when you put the pieces together. Alphabet and Number puzzles are also great for toddlers and as they get older too. They help with letter and number recognition and you can practice letter sounds with them. We have an alphabet puzzle that says, "A is for Apple, B is for..." as you complete that letter. My son loves it and at 2 years old, is already becoming familiar with recognizing letters and letter sounds.
The clock puzzle is fun to teach time (although time is a grade 3 skill that you learn, around the age of 7-9). The construction set and large floor puzzles are fun as your toddler gets older too! Floor puzzles create interest because they are so big. The pieces are also easier for younger children to put together.
As your children grow older, they will like more complicated puzzles. 3D puzzles are amazing. I have even found some 2D puzzles that also convert into 3D puzzles that are appropriate for ages 3 and up. Some can be painted, some are already decorated. Puzzles with more and more pieces are great as kids get older. There are so many puzzles with different images that teach different things that will engage your children. If they are learning about different countries and continents, you can get them a puzzle with pieces that teach that. If you are learning about planets or your kids are interested in it, solar system puzzles may be of interest. There is such a wide range that anyone, even adults can do them! Puzzles help engage the brain in a way that could even prevent dementia as we age (https://www.nia.nih.gov/alzheimers/publication/preventing-alzheimers-disease/search-alzheimers-prevention-strategies