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Talk to your friends, relatives, neighbors, classmates, and coworkers about what sorts of books they like to read. Gauging demand will help you tailor your selection to the needs, interests, and tastes of the people who will ultimately be making use of it. Think about the people that make up your community. If they're mostly retirees, you might make room for more large-print books and periodicals. If there are quite a few families, a well-stocked children's section is likely to be a hit. If space allows, you also have the option of carrying a wide variety of genres and titles to ensure that there's something there for everyone. Browse used book stores, consignment shops, flea markets, and garage sales for selections that you think would make good additions to your library. This is perhaps the best way to put together the core of your collection, as you'll pay little to no money for most of the books you find but still have the freedom to be as choosy as you like. Online book sellers like Amazon, Better World Books, AbeBooks, and Half.com also typically have extensive selections of used books at low prices. Only buy books that are in decent shape, as you can expect them to pick up a little wear and tear once they start changing hands. Pass on titles with torn or faded covers, loose or broken binding, missing pages, water damage, or heavily stained or soiled spots. Create a social media account for your library that you can use to spread the word about your need for donations. You could also distribute flyers or rely on good ol' word-of-mouth campaigning. Provide detailed information about the kinds of books you're looking for, as well as the general condition they need to be in. Hold an on-site book drive event where people can come to unload their old and unwanted books, or start a mobile collection service to go around and pick them up yourself. Keep in mind that many of the donations you receive will be things that people are looking to get rid of, which means they may not all be titles that your members are dying to read. Get in touch with the marketing departments of various publishing houses and let them know that you're starting a library and would like to feature some of their titles. Many companies are happy to negotiate deals to supply library owners with wholesale quantities of books at special discounted prices. Most publishing houses provide contact info for marketing and business-related inquiries on their websites. Make it clear to the representative you speak to that you don't intend to sell their books for profit. Otherwise, they may try to charge you a higher distributor rate.

Summary:
Find out what kind of demand there is for certain books in your area. Score previously-owned books through second hand sources. Solicit donations from the members of your community. Make a deal with publishers to get distribution rights for new releases.