As you probably know, one of the basic tenets of good creative writing is to vary word choice and sentence structure to keep from putting the reader to sleep, if nothing else. But this idea is also good practice for the start of any story. If all your stories begin in much the same way, you might want to mix things up a little.
For example, do all your stories start with your hero riding down a dusty road for one reason or another? Or perhaps you are using the variation of the same basic opening. For example, in story 1, the main character is trying to join an organization but can't; in story 2, the character belongs to a guild, church, or cabal, or other group and wants to advance but cannot. While the story can go anywhere from this common beginning, why repeat yourself in this way? More importantly, by using a common beginning, you set up an expection in the reader that the current story will be much like the last one and who wants to reader the same story twice?
Rather you need to try and find a unique opening. For example, if your hero is riding down a dusty road in one story then don't have him or her do that again. Next time open in a tavern drinking with his friends or camping out in the wilderness or engaged in a heated argument or waking up in the morning. Anything but the opening you already used.
The point here is that your opening matters... a lot. It determined whether the reader will continue to the end or stop before the story gets going. It is up to you to enthrall the reader to the point where he or she cannot turn away. When that does not happen, your story fails to do its job. When it works, you've got a winner.
How do you do that? There are lots of ways but the main thing is you need a hook. The hook is something to engage the reader, like a mystery or some conflict that will be resolved by the end of the story. Example: in one of my stories, the main character finds a baby in the ruins he is clearing out. The story ends when the readers learns who the baby is and what she was doing there.
Example: The main character wants to unveil a statue to honor the victims of a horrible attack on the town. The resident ghost doesn't like this idea and plans stop the proceedings. The story ends when the ghost is defeated.
So pay attention to how you start your stories. Mix it up a little and have fun with it. And by all means, don't repeat yourself.