Here are some tips I try to follow:
Make me care about the character. It won't matter to me if a character is in life-or-death peril if I don't care about her yet. If I invest my time in a story--if I keep turning pages--it's because I connect with the character, THEN through her the story and the world. Warning: this doesn't mean claustrophobic pages of internal monologue, because you must...
Start with story; start in-scene. Don't write about what has happened. Write about what's happening NOW. Nix those prologues, chunks of backstory, or lengthy interiority. Stick in the now.
Give a little framework before jumping into dialogue. Even a couple sentences will suffice. Give those talking heads some setting or sensory detail to help ground the reader.
Sprinkle in worldbuilding. I don't want to be inundated with the details of your world. I want just enough to ground me and intrigue me, but the story must keep moving forward. Remember when you're writing in-scene, beats must happen in real time. Narrative summary, lengthy descriptions, or lengthy interior reflection shouldn't break up in-scene moments unless there is enough "real time" allotted for it. Save narrative summary for transitions between scenes, if possible. Trust that the reader will catch on, BUT don't be overly coy, thinking that will add suspense. (It won't; it will only add frustration.) Deciding when to write in-scene vs. narrative summary is tricky, but you can do it!
Incite me. There needs to be an inciting incident by the end of chapter one (especially if you're writing YA). This doesn't have to be the doorway between acts one and two, but something BIG needs to happen that's life-changing for your character, that incites her into some kind of action. In other words, get your story wheels turning ASAP and make me NEED to read chapter two!
Begin on the day something different happens. Carol Lynch Williams gave this awesome advice at the WIFYR conference I just attended. Yes, you need to ground the reader in the "normal life" of your main character, but your story should start on the day life changes for her. For example, my story, The Rowaness of Shalott, begins when King Arthur comes to Guinevere's island after a five-year absence.
Conflict. Just because the main plot of your story hasn't kicked into full gear doesn't mean there should be no conflict at the beginning. There has to be conflict! Your main character must want something right from the get-go, and something/someone must be standing in her way; furthermore, there must be stakes if she doesn't succeed. (Hint: conflict/yearning is a key ingredient to making me care about your character.)
Don't forget the details. It's your job as a writer to not only engage me in your character (and his/her story), but also let me know his/her age, gender, physical appearance, and other important details ASAP. Age and gender, especially, must be indicated in some manner on page one.
Make a promise. Your reader should understand the kind of novel they're reading from the first page. If the book is a fantasy, something must strongly hint of that fantasy in the beginning. If you're writing horror, scare me. If your novel is a mystery, give me something to puzzle over--again, on page one!
Shake hands with your ending. Your beginning and ending should be fun house mirror reflections of each other. They need to resonate and come full circle. In order to know where to begin your story, think of where you will end it and let that spark some ideas. This is something that can be finessed with revisions, but I strongly believe you should know how your story will end before you begin it.
That about covers it. Easy, schmeezy, right? (WRONG!) And here comes my final piece of advice for writing story beginnings...
RELAX! Try not to panic over all the rules. Know them. Read and study them. But then turn a blind eye, let go, and have fun! Don't worry, those rules are probably lingering in the back of your brain and keeping you fairly on track. (And they'll be waiting with a vengeance when it's time for revisions!) But while drafting, let yourself fall in love with the seed of a story and the birth of a character. Don't judge them just yet. :-)
What are your favorite tips for writing beginnings?