For me the best thing about being an author is reading to a live audience. But I know a lot of authors are nervous about reading publicly so I thought I'd offer my advice to nail a live reading.
There are two main aspects to consider, the first is what you read and the second is how best to prepare.
Dealing with the latter first, the simple advice is rehearse. If I'm doing a piece I've performed before, I'll practice it twice every day for the week ahead of the reading. If it's a new piece, then I will do that for at least a fortnight ahead. The length of the reading slot can also effect these timings.
Why rehearse? Well there are three advantages I can think of:
1) Familiarising yourself so that when you come to read live there are no surprises in your own text that catch you out. This may sound a bit odd, after all it's your own text. But you'd be surprised, you may have written it quite a while ago because the publishing process can take a long time. Or just as significantly, when you were writing it, you weren't likely to be writing it with reading it out aloud and there are things that translate differently from page to voice. For example, I wrote a pun on 'greased lightning', with 'greased' written as the country Greece'd. Rehearsing I realised there was no possible way that this would come over to the audience and had to factor that in. Rehearsing and you may come across tongue twisters, difficult words to pronounce or alliteration that ties you up in knots, so practice and you can conquer them.
2) Which brings us to timing. Apart from a rehearsal allowing you to time the length of your reading if you have been given a time limit (which you always will in any open mic, but even often when you are on the bill as a named performer), rehearsing is vital to help you pace yourself. With the adrenaline running once you get up on stage to stand by the mic, it's the most natural thing to speed up and belt through your reading. Rehearsing maximises your chances for keeping the reading speed under control. The more measured the pace, the more chance the audience have of taking in your words. The more comfortable you feel up there, the more rehearsed, the less the tendency to tear through your piece.
3) Bringing the work alive. There is nothing worse than a reading which takes the audience back to listening to a dreary schoolteacher just reading to them from a textbook and making no eye contact. While you probably won't be making eye contact with individuals in the audience if the lights are low, it's still advisable to look up from the text and look out at the audience. It helps establish a genuine two way relationship and a rapport. I wrote the opening to a short story that went
"What is the ideal length for a suicide note? Asking for a friend".
When I read this story, I always look up from the text when I deliver the line 'asking for a friend'. It not only establishes a connection, it actually puts me and the audience together in a complicity - they realise that the character is not really asking on behalf of someone else, but trying to disguise the fact he's talking about himself. So the simple action of looking up actually helps establish the parameters of this story within its opening two lines. Again, I had no idea of any of that when I was writing the story, but through rehearsal its importance emerged.
It's not just about looking and connecting with the audience by sight. If you're feeling confident, you can enhance the story with gesture or an expression thrown out to them. You are to some extent acting out the story, albeit with one hand since the other is occupied holding the book. Rehearsal does not mean you have to learnt the text by heart to free both hands - poets can learn their poems because they have compression & specialist rhythm to help them master their work. It's not the same for prose writing, even though rhythm is important for us as well, it's not necessarily designed for projecting through voice, more structuring the reader's journey through the sentences. If you don't feel confident enough to do gestures and expression when you're starting out that's fine, but in time you're very likely to reach such a level.
Finally rehearsal allows you to accent certain words or phrases that again may enhance the meaning of the piece. I don't have much variety in my voice and can't do other accents, but you might have this ability which may contribute to your work aloud. It's because I don't have a terribly interesting voice that I adopt a lot of gesture and performance in my readings to compensate.
How to choose what to read. This is a harder one to be definitive about. First it depends what options you have. In all likelihood you are talking about your debut book, so you're restricted to that. Especially if the reading is mainly aimed at promoting that book. I'm going to assume it's a novel, since short stories and flash fiction are much easier to do live in that unlike an extract from a novel, with these the audience are getting the whole piece and therefore require less contextualising. For the past 4 years I have mainly been reading my flash fiction, but with a new novel coming out this summer, I'm going to be switching back to reading extracts and so what follows applies to me as I make my selections in the next few months.
Firstly what type of book have you written? Action thriller, literary fiction, romance, horror? This will inform your thinking as behind what you choose to read. What are you hoping to convey with what you read? Are you after conveying the style of the book? Or give a taste of the main characters and their relationship? Do you want to convey the atmosphere of the book (such as in Horror or Supernatural)? Do you want to read something that ramps up the tension in the room, then leave the audience on a cliffhanger? Maybe you want to make the audience laugh, or perhaps present yourself as a storyteller par excellence. All of these are valid but always derive from the book you have written. If you have a long reading slot of course you might be able to present a couple of different ones of these impressions. So these are the thoughts you ought to consider when choosing your piece. A descriptive piece may convey the atmosphere and the style of writing, but the downside is it may be hard for the audience to picture in their minds coming to it cold. Dialogue heavy extract may best convey the relationships and characters, but can you carry off the different voices to make them distinctive enough to the ear no matter how distinctive they are on the page.
If you can read the opening of your novel it is always useful as it is doing the job of providing context, rather than if your first extract is somewhere further inside the novel when you will almost certainly have to offer a preamble of how the plot got to that point. A general rule is less explanatory preamble and more reading of the actual text if you can possibly manage it. The preamble, or bits in between the extracts are your chance to talk about the book as a whole and do a selling job on it, sort of pitching it subtly and the extracts will hopefully show that off to the best manner. In the bits between the extracts you can talk about all manner of things associated with the novel, such as where you got the idea from, or which writers inspired you, these are nearly always more engaging for an audience than them trying to grasp your novel summed up in a few sentences. When I come to selecting my pieces (and I won't be selecting the opening of the novel), I may stick with my flash fiction approach and give minimal 1-line preamble and trust to the extract itself doing all the work and conveying to the audience what it's supposed to convey. The novel's structure has the advantage that it is episodic and those episodes occur out of sequence within the book itself, so they are fairly self-contained chapters. Having said that, one of the three main voices I can't read aloud at all, since there are visual cues and ideograms in the book that I just couldn't reproduce in a reading.
So there you have it. I wish you all luck with your live readings and please feel free to ask me any questions in the comments and I'll answer them.