Learn French: Le manège est si rapide qu'il me secoue. - The ride is so fast that it shakes me.

Yes! Both si and tellement can be used before an adjective like rapide to mean "so". You could absolutely say Le manège est tellement rapide qu'il me secoue. si is slightly more common in written French, while tellement is very popular in everyday speech.

Could I say si vite instead of si rapide here?

No, because rapide is an adjective describing the noun Le manège (The ride is fast). vite is an adverb used to describe an action. If the sentence used an action verb like "goes" instead of the verb "is", you would use vite: Le manège va si vite qu'il me secoue (The ride goes so fast that it shakes me).

I thought a manège was for horses. Why does it translate to "ride"?

You're right that traditionally, a manège is a horse-riding arena or riding school. However, in modern everyday French, un manège is the standard word for a carousel, a merry-go-round, or almost any amusement park ride.

Why is it il me secoue instead of il secoue moi?