Learn German: Die Reise war sehr teuer. - The trip was very expensive.

You can say that, and it is perfectly correct grammar! However, Germans almost always use the simple past tense (war) for the verbs sein (to be) and haben (to have), even in casual everyday speech where the Perfekt tense is normally used for everything else. It is much more common to say Die Reise war... than Die Reise ist ... gewesen simply because it is shorter and easier to say.

Die Reise is a feminine noun. Why doesn't teuer get a feminine ending to match?

Adjectives in German only get gender endings when they are placed directly before a noun (as in die teure Reise - the expensive trip). When the adjective comes after a verb like sein (like "the trip was expensive"), it is called a predicate adjective and stays in its basic, unchanged dictionary form (teuer).

Why is the verb form exactly war here?

War is the simple past tense of sein (to be). Just like in English (I was, you were), the verb changes depending on the subject. For ich (I) and the third-person singular er/sie/es (he/she/it), the form is war. Because die Reise (the trip) acts as a "she" or "it" in grammatical terms, it takes the war form.