Learn Russian: Я ужасно устал после дороги.

In Russian, getting tired is viewed as an action that has already been completed. Я устал literally means "I have gotten tired", which describes your current state resulting from that action. If you used the present tense я устаю, it would mean "I get tired" on a regular basis, rather than how you feel right now.

Doesn't ужасно mean "terribly" in a bad way? Can it just mean "very"?

Yes, ужасно comes from ужас (horror), but just like "terribly" or "awfully" in English, it is frequently used as a strong intensifier in everyday speech. Here, it simply means you are extremely tired, without necessarily implying a literal sense of horror.

I learned that дорога means "road". Why is it translated as "journey" here?

While дорога literally means "road," it is very commonly used to refer to the time spent traveling or the journey itself. When someone says после дороги, they mean "after being on the road" or "after the trip," especially when referring to the fatigue of a long drive or train ride.

Why is the word дороги used instead of the dictionary form дорога?

The preposition после (after) always requires the genitive case. Since дорога is a feminine noun ending in -а, its genitive form changes the ending to -и, becoming дороги.