I really only try to learn songs from anime and games right now since I know I'm going to hear them all the time. Like right now I'm trying to memorize this song 言わないけどね (I won't say it, but (right?)) from this anime called からかい上手の高木さん (Karakai Jouzu no Takagi-san, called Teasing Master Takagi-san in the US) that I have been watching lately. It's a little above my level since it's about 12-13 year-olds and I do better with stuff like しろくまカフェ (Shirokuma Cafe, or Polar Bear Cafe in English) targeted at say 9 year-olds. But it's funny and the writing is good. Here's the song:
Though I'll occasionally try to learn a song unrelated to anime / games if it's kind of easy and famous, like this huge hit from the 1960s called 上を向いて歩こう = ue wo muite arukou = look up when you walk.