It has been a while since I studied it, but I've used this as a course text to a course I couldn't attend the lectures of, and it wasn't exceptionally hard to pass. Your comments give no immediate reason to suspect it would be a bad choice; but be prepared for a quite substantial amount of proofs "left to the reader".
Euler apparently had some trouble deriving the Jacobian used in change of variables for double integrals.
He began by considering congruent transformations consisting of (affine) linear functions, and got something like $$\mathrm{d}x\,\mathrm{d}y=m\sqrt{1-m^2}\,\mathrm{d}t^2+(1-2m^2)\,\mathrm{d}t\,\mathrm{d}v-m\sqrt{1-m^2}\,\mathrm{d}v^2$$ which he described as "obviously wrong and even meaningless." He then got
$$\mathrm{d}x\,\mathrm{d}y=\left(\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}\right)\,\mathrm{d}t\,\mathrm{d}v$$ which was not symmetric in the variables, and therefore would not do. Finally, he derived the correct
$$\mathrm{d}x\,\mathrm{d}y=\left|\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}-\frac{\partial y}{\partial t}\frac{\partial x}{\partial v}\right|\,\mathrm{d}t\,\mathrm{d}v$$ and lamented that simply multiplying out $$\mathrm{d}x\,\mathrm{d}y=\left(\frac{\partial x}{\partial t}\,\mathrm{d}t+\frac{\partial x}{\partial v}\,\mathrm{d}v\right)\left(\frac{\partial y}{\partial t}\,\mathrm{d}t+\frac{\partial y}{\partial v}\,\mathrm{d}v\right)=\left|\frac{\partial y}{\partial v}\frac{\partial x}{\partial t}+\frac{\partial y}{\partial t}\frac{\partial x}{\partial v}\right|\,\mathrm{d}t\,\mathrm{d}v$$ and shredding the squared differentials yielded an incorrect but annoyingly close answer.
But let us remember, if Euler committed errors it was only because of the unrivaled breadth of his work. If I could finish with a quote from the article cited below: "As a developer of algorithms to solve problems of various sorts, Euler has never been surpassed."
Source: For an excellent review of the history of the Jacobian, and to learn more about the details of what I have written, I highly recommend reading this article by Prof. Victor J. Katz (Internet Archive, jstor.
Best Answer
I have studied Euler's book firsthand (I suspect unlike some of the editors who left comments above) and found it to be a wonderful and illuminating book, in line with Weil's comments. You will gain from it a deeper understanding of analysis than from modern textbooks. It is true that Euler did not work with the derivative but he worked with the ratio of vanishing quantities (a.k.a. infinitesimals), which actually turns out to be a more general concept, but this is a subject for another post.
Note: we just published a detailed study of Euler that hopefully sets the record straight and vindicates Weil's hunch.