Why not take a look on the sections on model theory in the reading Guide on math logic which you can download from http://www.logicmatters.net/tyl ?
Yes, Marker's text is standard; but it is tough. The Guide suggests a few texts that bridge the gap between the entry-level model theory in standard first math logic courses and the advanced text by Marker. The headline news is that Hodges's Shorter Model Theory is a particularly good intermediate choice: but there is a lot more said in the Guide.
I learned multivariable calculus from Paul's Online Math Notes.
If you want a physical textbook, I second Jared's recommendation of Marsden & Tromba's Vector Calculus. It has a somewhat more theoretical flavor to it than James Stewart's books.
Another standard text is Edwards & Penney, which I've used to tutor students. However, it's essentially on the same plane as Stewart.
Now for a few comments.
First of all, if you're studying for the GRE, then you might not want a textbook that emphasizes theory. First and foremost, you need to be able to solve basic problems and calculate things, so in that sense a book like Stewart's might actually be the most appropriate.
Speaking of Stewart, not everyone holds his books in such disregard. I don't love his textbooks personally, but I do understand and appreciate why they're the standard.
Finally, I'd like to take a second and exude some enthusiasm for the subject. Multivariable calculus is one of my favorite areas of math, and was crucial in helping me develop intuition for (and interest in) differential geometry. In my (admittedly limited) experience, undergraduates skipping multivariable calculus and ordinary differential equations is not too atypical. However, I would hope that all serious math students eventually go back and learn both subjects, appreciating them for their inherent beauty.
Best Answer
Take a look at my Teach Yourself Logic Study Guide, which gives a lot of detailed advice about logic books at different levels, suitable to different backgrounds.
Check out the proposed syllabus of the course you are about to start to see what it covers and hence what's relevant in the Guide.