First, Do you really need an agent? Yes, if you want to try the big publishing houses. But many smaller and niche publishers are happy to get submissions directly from the author. If you do get an offer from a publisher, it can be a good idea to then hire an agent or lawyer specializing in publishing to read the contract and let you know what exactly you are signing up for. As with the self-publishing decision, the "to agent or not to agent" question requires you to be realistic with your goals for your book.
The traditional way to find an agent. Ask around; ask other authors about their agents. Are they accepting new clients? What kinds of books do they rep? What books have they sold? Do they return your phone calls? Do you like working with them? Read the Acknowledgements in similar books: Authors usually thank their agents, as well as their editors and publishers. Finding books like yours is also a good way to find appropriate publishers. Research: Check out Poets & Writers, Writer's Digest, other mags and sites, and do an internet search on "finding legitimate literary agent" or some such. There are enough guides out there for finding agents, so I won't cover ground that's been covered, except to say...
Don't get scammed. Unlike lawyering, school-teaching, motorcycle driving and maybe even haircutting, literary agenting requires no certification. I could hang out a shingle--which these days means starting a web site--and start calling myself a literary agent today. (Hmmm...perhaps not a bad idea...). So make sure they've actually sold some books and have had a publishers lunch or two. And if they ask you for money, put your transmission into reverse and back away. Yes, some legitimate agents will sometimes reject a manuscript saying it has promise but is not there yet, and then recommend manuscript editors you can hire to help you get it there. I've been one of those recommended editors; we can be a big help, and if the agent recommends us, you know they trust out work. These days, some agents (and even publishers) don't have the time or resources to take a decent manuscript or proposal and help edit it into a great one, so independent editors can be crucial. But think of all the ways this situation can be exploited. Way # 1: Agents promise to look at your manuscript rewrite if you hire a certain editor, who happens to work for them, to re-shape it. Most legitimate agents work on commission, not upfront fees for editing or anything else; they get paid when they sell your book.
Everyone complains about their agent. Some agents put a lot of effort into editing and crafting your manuscript and book proposal. Others expect you to do the writing, while they focus on the selling. Some are excellent deal-makers and can negotiate a great deal with terms you didn't even know existed. Others are known for their extensive contacts--maybe they used to be publishers or they take a lot of publishers to lunch--and they know who's buying what right now. It's almost unheard of to get a super-hero super-agent who can do it all.
Follow the etiquette. Again, plenty of books and internet sites have covered the step-by-step protocol of sending agent query letters and book proposals, so I won't. When I wrote my first book proposal, my agent told me to follow the guidelines in The Shortest Distance Between You and a Published Book, so I did. It's old, but it's a great how-to. There are other good guides out there, too. For insider information on the publishing business, Pat Walsh's 78 Reasons Why Your Book May Never be Published and 14 Reasons Why It Just Might is entertaining and informative, though it does NOT offer a step-by-step guide to constructing a book proposal or query letter. (Walsh, full-disclosure,is the publisher/editor of my memoir To Have Not.) Suffice it to say you do not want your first communication to a prospective agent to be an email with your entire 600-page manuscript attached.
How ready should your book be? If it's fiction or a memoir, it should be finished before querying an agent or publisher. Nonfiction has traditionally been sold based on the first three chapters and a detailed proposal, which includes the Table of Contents (TOC), a chapter-by-chapter synopsis, and more. But with first-time authors, agents and publishers may want to see more finished chapters. You can't blame them, really, for wanting more insurance that a) you'll finish the book; b) they're getting a book they want; and c) it'll be a good book.
Will I put in a good word for you with my agent? Probably not. If I put in a good word for everyone who asked me to, my word wouldn't count for much with my agent, now would it? Over the years, I think I've recommended three names to her, and these people had books that I thought were juuuust right for my agent--and she still didn't take any of them. So even a good word may not do much.