The second game pretty much assumes that you know everything that happens in the first game. That includes all the major plot twists, how the characters develop, and even what happens in the side quests (depending on if you did them or not). You could jump into the second game if you want to but you will be completely lost on what is going on.
I just got the game a few days ago and took some time to make a level. It is a simple level design-wise that involves a lot of Goombas and Pirahna Plants. (C2F6-0000-00DB-1F6B) (The embed function of the Miiverse post does not seem to be working for me)
I just beat the final boss and I have to say that I LOVE the game! It came really close to matching the quality of the original game. I have read about people complaining about the recycling of bosses. What actually happens is that while the game does recycle boss characters, the fights themselves are different enough. I actually appreciate that. The game starts out simple enough but as the game progresses, the levels have their own gimmicks that do not deviate from the formula but offer cool new ideas that the original game did not have. Some of my favorites come from the haunted house levels, especially later in the game when it presented a new utility for the Boos which I was pleasantly surprised by. The one downside I noticed is that checkpoints in boss levels are not always placed next to the boss door, something that the original always had. Overall, I would say that of the games that have come out this year, Yoshi's Woolly World is a close contender for my Game of the Year. (doubt it will beat Splatoon though)