Complete your book before approaching an agent or editor. It's easier for an agent to sell your manuscript to publishers like Random House when they can show the editors a finished product.
Make a list of literary agents that might be suitable representatives of your work. Use a directory like "The Writers' and Artists' Yearbook" to look through the types of books and authors that each agent normally handles and make a list of the most appropriate options. Send your work to them in order of preference, and be aware that it usually takes weeks or months for an agent (or publisher) to reply.
Write a cover letter, or query, to send to the agent along with your manuscript. This should state very briefly what your book is about, the genre (young adult, fantasy, literary, chick-lit, non-fiction), the word count (70,000 words for a standard literary novel, less for children's or young adult books, more for fantasy, while non-fiction varies) and some information about yourself. Keep this information relevant, including information like previous publications rather than biographical information about you or your family.
Write a synopsis of your book to send with your manuscript. This is a two-page summary of the events of contents of your book from beginning to end. This allows the agent to get an overview of your book, so that she can decide whether or not to offer you a contract.
Post the first three chapters of your manuscript to an agent, along with your query letter and synopsis, making sure everything is printed and double-spaced, with page numbers on each sheet. If you're not sure what to do, contact the agent's office to check their submission requirements. Many agents don't accept manuscripts via email attachment. Don't sent handwritten scripts--agents have to sift through hundreds of manuscripts each week and the easier they are to read, the better. Most agents don't want the entire manuscript at this stage--again, check the individual agent's requirements.
Instruct your chosen agent, when you get an offer of representation, that you want to get published by Random House. She will know which particular editors to target and the best way to approach them.