Learn Spanish (Spain): Llevaré el papel al contenedor.

In Spanish, llevar means 'to take' in the sense of transporting or carrying something from one place to another. Tomar means 'to take' in the sense of physically grabbing something, catching a bus, or having a drink. Since you are moving the paper to the bin, llevar is the correct choice.

Could I use traer (to bring) instead of llevar in this context?

No, because of the direction of movement. Llevar is used when taking something away from your current location to somewhere else. Traer is used when bringing something towards your current location. Since you are taking the paper away to the bin, you must use the future tense form llevaré.

Is contenedor the standard word for a trash bin in Spain?

Yes, but specifically for large street bins or dumpsters, like the big blue public recycling bins for paper (el contenedor azul). If you were talking about a small wastebasket under a desk or inside a house, you would use the word la papelera instead.

Why does the sentence use el papel (the paper) if English might just say 'I will take paper to the bin'?

In Spanish, when you talk about a generic substance or material, you usually need to include the definite article (el or la). Since this sentence likely refers to recycling paper as a general category of waste, you include the article and say el papel.