So I recently got a flyer saying I can apparently order 20 free analog ICs from Texas Instruments. Being the newby that I am, I honestly don't know what I should get.
Around here there's no such thing as free, and flyers like that always have some kind of catch (like got to buy something else at inflated prices or sign up for something etc).
But if this deal is legit in your case then I'd start with a couple of things that are useful for many different projects. Linear regulators and op-amps would probably top the list of things I like to have on hand as basic parts for simple projects.