Well, through the 50s, I was pretty much stuck with my parents' record collection which was mainly obscure classical stuff. However, somebody had given them a Mills Brothers record one Christmas and I recall playing "Get a Job" and "Yellow Bird" over and over.
Things changed in the early 1960s when I saved up for months and bought myself a....TRANSISTOR RADIO! My first discoveries were some novelty songs--"Monster Mash" standing out big time and also "Hello Muddah, Hello Fadduh". They don't write them like that any more!
However, I'm ashamed to admit that the first song I liked well enough to actually spend a dollar for a 45rpm copy of was "Downtown" by Petula Clark.