I see だけど used frequently in conversations, whereas しかし is predominantly used on TV programs. Thank you.
しかし is more formal. It`s often used at the beginning of a sentence, しかし is like "however." だけど is like "but."
"However," is definitely more formal than "but." Think of Shikashi and Dakedo in the same way :-)
My wife is Japanese, and she told me this.