Learn Russian: Завтра мы будем писать слова этими фломастерами. - Tomorrow we will write words with these markers.

In Russian, when you use an object as a tool to do something (like writing with a marker or eating with a fork), you just put that object in the Instrumental case. You do not use the preposition с here, because that would mean writing 'together with' the markers as if they were your friends.

How do the endings work for этими фломастерами?

The base word is фломастер, a masculine noun ending in a hard consonant. To make it Instrumental plural, we add the standard ending -ами to get фломастерами. The word этими is the Instrumental plural form of эти (these), which changes to match the noun it describes.

Why does this sentence use будем писать instead of a single future tense verb?

Because писать is an imperfective verb, meaning it focuses on the ongoing process of writing. To put an imperfective verb in the future tense, you must use a helper verb (будем for 'we will') plus the infinitive. If the focus was on completing the words, we would use a single perfective verb like напишем.

What case is слова in here? It looks like the singular Genitive.

It is in the Accusative plural. The base word is the neuter noun слово, and its plural form is слова. Since words are inanimate (not alive), their Accusative plural form is exactly the same as their Nominative plural form. It just happens to look identical to the singular Genitive.