In your child's second year, these things might be useful or interesting. They were for us.
Melissa and Doug -
They make quality wooden toys perfect for this age group, and can usually be found on sale on Amazon. This car carrier has been a hit with every kid I've ever met. They also have play sets with a vehicle and several figurines. The tractor and airplane were most popular at our house.
Chunky puzzles are great simple shapes that also have a picture of the piece for easier matching. The pieces also stand up on their own, for creative play.
Learning Play -
This is the time where kiddos begin to mimic. So pint-size everyday items can be big hits. This broom and mop set is great. A toy stroller or wagon to push and pull stuffed friends. For outside, a garden set is perfect for digging in the dirt or sand.
Active toys -
A pop-up tunnel is good for outside or the living room on a rainy day.
Little Tikes makes great ride-ons. Our daycare had this one and it was fabulous. At home we had the Prince Lionheart Wheely Bug, and it was fun, but is more difficult because of its construction.
Now that they can somewhat feed themselves, these snack catchers are great - especially for the stroller or car. Out to eat at a restaurant can be challenging, so we always took our own plasticware (these have a carry case!) and a waterbottle (since some places don't have cups with straws/lids).