I’m shopping at the grocery store.
As for shopping, I usually just do it online.
We’re going Christmas shopping today! I’m so excited!
You've hopefully seen 買う by now and know it means "to buy." So, 買い is going to be similar to that. Add that to thing and you have a "buying things" sort of word. What does "buying things" describe? It describes shopping.
The readings are the readings you learned with the kanji. Also, you can use the reading for 買う to help you with 買い as well. 物 is just もの, like you learned.