(A) 店（みせ）を開（あ）ける。
(B) 店（みせ）を開（ひら）く。
Both sentences are correct and both can be translated as “I open a shop” but the meanings are usually not the same.
Sentence A is usually used at the beginning of each business day with the intention of closing locked at the end of the day, although very occasionally it is used to mean the same as Sentence B. On the other hand, Sentence B means to start a business. In this meaning, people also say 店（みせ）を出（だ）す.
In general, あける is used when it is clear where to start to open it and you know how to close it again. For example:
カバンをあける = to open a bag
窓（まど）をあけてください。= Please open the window.
On the other hand, ひらく is more like “to unfold” or more metaphorical. For example:
扇（おおぎ）をひらく = to open a fan心（こころ）の窓（まど）をひらく = to open up a window of the mind一般（いっぱん）の人（ひと）にも門（もん）をひらく = to accept general public
