Learn German: Ich werde an der Rezeption fragen.

This sentence is in the future tense. In German, when you use a helping verb like werden (will), it takes the second position in the sentence. The main action verb (fragen) is pushed to the very end in its dictionary form (infinitive).

Why does it say der Rezeption if Rezeption is a feminine noun?

You are seeing the dative case in action! The word an is a two-way preposition. Because you are talking about a fixed location (where you are asking) rather than movement to a new place, an triggers the dative case. In the dative, the feminine article die changes to der.

How would I use this lesson's topic, was für ein- (what kind of), to ask: At what kind of reception will I ask?

You would say: An was für einer Rezeption werde ich fragen? Because the preposition an requires the dative case here, and Rezeption is feminine, the ending for ein becomes einer.

Could I use the preposition bei instead of an here?

Yes, you can also say bei der Rezeption. Using an emphasizes being right up against the physical reception desk, while bei generally means being in the general area of the reception staff. Both sound completely natural.