Learn Spanish (Latin America): Ella lava los vasos cuidadosamente.

To make an adverb like carefully in Spanish, you take the feminine form of the adjective and add -mente. The adjective for careful is cuidadoso, so we change it to the feminine cuidadosa and add -mente to get cuidadosamente.

Does vasos mean reading glasses or drinking glasses?

Vasos specifically means drinking glasses. If she were washing her reading glasses, you would use a word like lentes or anteojos.

Could I put cuidadosamente right after the verb, like Ella lava cuidadosamente los vasos?

Yes, both are grammatically correct! Adverbs of manner (words telling you how something is done) can usually go right after the verb or at the very end of the sentence. Putting it at the end often sounds a bit more natural when the object (los vasos) is short.

Is it necessary to include Ella at the beginning?

It is not strictly required because the verb lava already tells us it is a third-person subject. However, since lava could mean he washes, she washes, or you (formal) wash, keeping Ella makes it perfectly clear who is doing the washing.