Learn Romanian: Noi vedem treisprezece muzee.

They are built using a simple formula: the base number + spre (towards) + zece (ten). So treisprezece is literally "three towards ten" (trei + spre + zece). You will see this exact same pattern for all numbers in the teens!

Why isn't there a de between the number and the noun here?

In Romanian, you only need to add the word de (of) between a number and a noun if the number is 20 or higher. For numbers 1 to 19, the number goes directly next to the noun, just like in English.

Why does the word for museum (muzeu) become muzee in the plural?

Muzeu is a neuter noun. Neuter nouns that end in -eu in the singular usually drop the -u and add an -e to make the plural. This gives them that double e at the end. Another common example is liceu (high school) which becomes licee.

Do I always have to say noi before vedem?

No, you can safely leave it out and just say Vedem treisprezece muzee. Because the verb changes its shape to match the subject, the ending of vedem already makes it perfectly clear that "we" are the ones seeing.