Sounds like your dog could be alert barking. In their mind, they think: “Mom- a car! Mom- a cat! Mom- a leaf!” Your dog is likely not barking at daycare because they are having too much fun.
Even though you take them to daycare, they may need more exercise or may need to go more. Tired dogs are less likely to bark in the house.
Teach them that they don’t need to tell you about everything they hear/see. To do this, start with things you can control, such as person walking by the house. Ask a friend or family member to help. Have the person walk by. As soon as they see the friend, IMMEDIATELY start praising them and give them a treat. If you miss the moment and they start barking, just wait for them to be quiet and try again. If you start yelling at them to stop barking (NO, NO, DON’T BARK!) then they will only think you are barking with them!
Be sure to practice obedience commands. If we want a dog to listen to us when they are reacting or barking, then they also have to listen when they are in the house or on a walk. Make them sit, lie down, and stay multiple times a day just because you said they should.