I haven’t read Yotsubato specifically, but judging from my own early experiences of reading manga (I started with Takagi-san), it was mostly the casual contractions that tripped me up. The good news is that after encountering them a number of times they become second nature. The other good news is that there are lists of such contractions you can consult. One is even here on this forum:
And never hesitate to ask, either in the relevant thread, or the grammar thread, or by making a new thread of your own. Having something personally explained to you sometimes helps more than reading lists of grammar points. And be reassured, whatever someone’s level when they start reading native content, their experience will always be similar to yours. Mine certainly was. 