Olá, @mdomocol. In any context, the difference between levar and trazer is comparable to the difference between to take and to bring. There is a specific logic behind each verb, which is not to say that people don’t switch it up sometimes in the real world 
We can think of levar (to take) as divergent – suggesting a movement from here to there – and trazer (to bring) as convergent – suggesting a movement from there to here. So, even when it seems that there is no practical difference, the nuance is there.
- Levarei as crianças à escola → Applicable, for example, when you’re home and telling this to your spouse next to you. Home is your here and you’ll be moving away from it to go to the school, which is there for both of you.
- Trarei as crianças à escola → Applicable, for example, when you’re on the phone with a teacher waiting for your kids at school. You’ll be moving towards the teacher, which is already here for your listener and will eventually become here for both of you.
Between levantar and buscar, in the sense of picking something up, the differences are the following:
- Levantar → Used for ‘formal’ collection processes, such as collecting an order in a store, collecting a boarding pass at the airport, etc. Not applicable to people or animals.
- Buscar → Used much more broadly. So, it can be used to refer to picking up people, getting something from your room, etc.
Considering the above, only your second example is idiomatic:
Vou buscar as crianças à escola.