The first is free-hand, which is to say that you record piano and vocals at the same time and even if they’re off tempo the two are synced together. It will sound more free and raw, but you’ll have a hard time syncing rhythmic elements and timed processing such as delay and reverb in a consistent manner. The second way is to record on grid, whereby you’ll record to a click-track to steady your tempo. In this case, it’s best to record one track at a time, but that doesn’t mean you can’t sing to yourself while you record your keyboard takes.
Everyone can sing this tune off the top of their heads! Listen to “Frosty the Snowman,” and focus on when the lyrical syllables move from “fros” to “ty” at around 0:12.