My parents read to me very near religiously. They had kids books aplenty- the Night Kitchen, Where the Wild Things Are, The Girl Who Cried Flowers (with Tanith Lee!) and a book I wish I could find called (I think) the Muffin Muncher, which in retrospect was kind of a libertarian parable about how taxation strangles industry, told in the form of a muffin-eating dragon. Dad committed a fair bit of Shel Silverstein to memory- he can still do Sara Silvia Cynthia Stout. Mom read me all the Narnia books, plus the Hobbit and the Lord of the Rings (in fact, I distinctly remember being so upset about the part with Shelob that I snuck the book to school the next day and read all during class and lunch just because I couldn't believe that had really happened to Frodo. It was one of the last books read to me before I was reading on my own.)
Mom is probably most directly responsible for my love of epic fantasy, though I'm sure there was some sort of feedback loop that went on there. She gave me the Belgariad and when I spent those years reading Dragonlance and Forgotten Realms she read them with me and we'd talk about them. She never put me in a corner about being a genre fan- in fact she encouraged it so thoroughly I wasn't really aware for a long time that there might be a stigma about it. She's since been a very reliable resource for all sorts of fabulous fiction. Not to long ago she put Lonesome Dove by Larry McMurtry in my hand and said it had always been one of her favorites. Now it's one of my favorites too. I also read Year of the Flood and Oryx and Crake at her request.
My father is no slouch either. He's personally quite a good poet, and his favorites are The Brothers Karamazov (which was his grandmother's favorite when he was growing up), Pilgrim at Tinker Creek (Annie Dillard), and the marvelous ouvre of Kurt Vonnegut (his favorite is Cat's Cradle, but I have to go with Slaughterhouse Five). He's also a fan of pithy quotes and humorous essays. I remember for years growing up he'd have a quote of the week pinned inside his office both at work and at home. What I learned from my father and the way he read was to appreciate language, word play, the melody of speech, and proper timing. He can memorize and recite all sorts of things, gives a hell of a speech, and taught me The New Colossus by Emma Lazarus one day just for something to do.
My mother's wife, of all my family, probably reads the most. She gravitates toward women's and queer fiction especially, and more or less all the nonfiction she consumes is highly political. She gave me my first copy of Even Cowgirls Get the Blues. She also introduced me to Barbara Kingsolver, and I routinely raid her shelves where our interests overlap(or when I find myself at her house without a book of my own, which is how I read both Ape and Essence and Dogs of Babel). Currently I have her copy of Virginia Woolf's Orlando hostage. Our tastes don't always overlap- I tend to veer away from heart-rending tragedies and she doesn't generally go in for the speculative fiction (with the exception of Atwood who she reads both in genre and out of it- so far I've not had much patience with her non-genre novels, but she writes a fantastic short story). I've recently pushed Ursula Le Guin's Left Hand of Darkness at her. I hope it sticks. She dug Reading Lolita in Tehran.
My little brother is an extensive reader. His favorites are Hemmingway, Hunter S. Thompson, and Vonnegut (so he and Dad and I can have a pretty lively discussion when we get together). Recently, I've gotten him hooked on comic books, where we both love Millar and Ellis (I've tried to point him at Gail Simone, but her best stuff is generally DC, and he's a Marvel guy right now). He's my go-to for literary literature, as for a while that was all he would take in. He insisted I read the Rum Diaries and I liked it quite a lot. I got him turned on to Tom Robbins and I think he may have read more of him than I have now. I tried to sell him on Neal Stephenson, but it didn't take. I also keep pushing him at Nabokov, which I think he'll like better, even though he's more of a bare prose man (he says he never could get through Faulkner). He's also a Harry Potter fanatic.
One of my stepsisters read a good bit of my penny dreadful dungeons and dragons fiction back in the day when I was doing that. She's since moved on to be a prolific reader of romance novels, which the whole family teases her about, though to be fair I've borrowed a couple of her regency romances and enjoyed them thoroughly if not without my tongue firmly planted in my cheek. We don't necessarily have a lot in common in our reading, but I do try to stick up for her literary choices in my own backhanded sort of way. It doesn't have to be Shakespeare to be fun. Her son is currently reading Percy Jackson the Lightning Thief, and I'm doing my best to cheer him on at every opportunity.
My other two sisters read somewhat less, but we still talk books every now and again. One of them put me to shame by reading Octavia Butler before I did, and we trade recommendations on those intermittent occasions when it's just us. Her husband and I once had a marathon (and slightly competitive) science fiction book off in terms of who of import we'd read and who we had on the shelf just waiting when we got home.
I have no idea what it's like to try to come into this business when one hasn't grown up surrounded by books and people who love them. Books haven't really ever been a lonely hobby to me. They've been something I could share over snacks with the people I love. We've always been able to have conversations about what we thought the point was or how we didn't see it that way, but we could see where they were coming from.
I guess the best advice I can give at the end of this is: if you do have kids, read to them, and let them know you like to read. It will make a huge difference.