Translation:Where is the closest convenience store?
It would be more natural to say "一番近いコンビニに行きたいんですが" - literally "I want to go to the closest convenience store...". As I understand it, the phrase どこですか implies that you expect the listener to know the answer and help you. But if you just say, "I'm trying to get to X", people will both understand what you're really asking, and consider it less rude.
一番（いちばん） literally means number one （一: one, 番: number), so in this context, it's 'the number one close convenience store'. With すきな, it's 'the number one like' (literally).
So actually, いちばん can be used in lots of places, since it is just turning the adjective it's attached to, into the superlative form of the adjective.
I understand the context that it is being used in in this sentence, I was just wondering if this was common usage among native speakers. For some reason, I have the (probably wrong) impression that ichiban would be more commonly used for things that are being described favorably.