The first series I read completely independent were the Boxcar Children books. I remember everyone I knew at school absolutely LOVED them but I was always kind of "meh" about them (the characters would obsessively talk about food and that drove me crazy). I also use to get together with friends and read the Arthur books (the show was good too!) and we'd also read the Amelia's Notebook series.
When I got a little older I loved the Usborne Puzzle Adventure books and Nancy Drew (Nancy Drew and the Hidden Staircase was my favorite!). I also really liked the American Girl books, especially Addy (though I had the Samantha doll and my mother always said I looked like Molly). I think the Addy books were the best written in the series and it was also through those books that I learned about the Civil War. My all-time favorite book series, however, were the Dear America books. I remember my mother giving me Across the Wide and Lonesome Prairie (about the Oregon Trail) and I read it in one sitting. My favorites were The Winter of Red Snow (about Valley Forge) and Standing in the Light (a girl abducted by Indians). I also really liked A Line in the Sand (about the Alamo) and I remember feeling really rattled by So Far From Home (about the Lowell Mills and Irish immigrants) because it was the first time a character I had grown to like died in the book.
One special series I grew up with were the My Book House books by Olive Beaupre Miller. They were my grandmother's when she was a girl in the early 1930s. Whenever my mother and I would go visit her, I got to read them. Then a few years ago, she gave me the series which I now keep will all my dearest books.