We don't get a lot of library books because I don't want to have to try and keep the books sorted from our own. We easily have 250 board and picture books. Possibly many more. I have many of them memorized from having read them time after time. Most books have been read multiple times.
When there's something I want to introduce to the kids (or they are interested in a subject) I ask for book recommendations here and I go to amazon.com and do a search. I buy based on the reviews. Once you've chosen your books, a good place to actually search for them is www.allbookstores.com
as they search for the cheapest books at numerous sites.
We've been reading to our kids since they were a couple months old. First with fabric books and board books and then into more complicated board books and picture books. My son is 4.5 and we are just now into our first chapter book. Both of my kids have always loved reading. We do not have a reading routine, we just read throughout the day. I try and be very animated when I read.
(This is really important.) When the kids are really little I don't always read the words, just point to pictures. My daughter just turned 2 and she's not liking that so much, she wants me to read the words.
We let them take turns choosing books. Sometimes I'll easily read for 45 minutes at a time. The first thing they want to do in the morning is read. Then they want to do it throughout the day, then at night. We may have 6 reading sessions throughout the day. If there's a book I want to try and my son does his normal resistance to something new, I just tell him I want to read it. I sit down and start reading. Soon he's in my lap and often he ends up liking the book.
We also use the sonlight list. I pick and choose which books I think are appropriate to our family.
My recommendation is to turn off the screens then sit down and start reading out loud to yourself and read with gusto. The kids will come to you. Open up your lap and have fun. Tell them they can each choose a book (or three.)
Hand, Hand, Fingers, Thumb http://www.amazon.com/Fingers-Thumb-...2628874&sr=8-1
was my son's first favorite book. Even now he loves it. It's a musical book with a great beat. If you read it monotone it is a stupid book.
A great board book: http://www.amazon.com/Barnyard-Dance...628995&sr=1-19
I stomp my feet to a square dance beat and call this out like I was the fiddler myself.
Great pictures and wonderful rhyming story: http://www.amazon.com/Seven-Silly-Ea...2629046&sr=1-1
Good series: http://www.amazon.com/s/ref=nb_sb_ss...29_1445&fsc=-1
Great intro to classic art (there's a whole series) http://www.amazon.com/Touch-Art-Brus...2629133&sr=1-1
We got the Laura Ingalls Wilder picture books http://www.amazon.com/s/ref=nb_sb_ss....70_298&fsc=-1
When I wanted to try (again) to see if my son was ready for chapter books I got the classic Little house in the big woods. My son was resistant so I just sat down and started reading. Of course, the first chapter is all about hunting and slaughter and we are hard core vegetarians so I really had to edit as I read. However, the second chapter was the foundation for http://www.amazon.com/Winter-Days-Bi...629434&sr=1-14
I made a huge deal about how this was all in the other book. I got out the picture book and we compared sections. He was hook, line, and sinkered into the chapter book.
My son really likes the Richard Scarry books http://www.amazon.com/s/ref=nb_sb_ss....155_148&fsc=4
He loves to look at the pictures. These were written awhile back so can be outdated in content. Like the child that goes to the hospital to have their tonsils taken out and the mom can't stay with the child. I tell him that's not how it is any more and he's okay with it. Try "what people do all day" http://www.amazon.com/Richard-Scarry...2630064&sr=1-1
or the please and thank you book.
So, again, just read out loud with great animation and the kids will come to you.
ETA: I can recommend a ton and a bunch of books. Tell me what your kids are interested in or what you would like to introduce them to and I'll see if I can recommend anything. Most of the books I referenced above can be bought for a penny plus $3.99 shipping used at amazon. Look at their used price box. If there's a choice to "Show # more formats" choose that to see what other options are available.