The Preliminary Exam for the Group in Logic & the Methodology of Science
In what has become a time-honored tradition (see Arc and Michael's thoughts), I have a few bits of advice on preparing for the mathematics prelim. My general sentiments on textbooks mirror those expressed by Michael. In my experience, Marker's text is the most cogent option; since the course has historically used Hodges', I would go so far as to say that you should read through Marker's text in parallel (Hodges occasionally takes a simple idea and makes it horrendously complex, e.g. his discussion of E-M models). For computability theory, I found it best to just read Soare's text from the first page. I've also posted my notes covering both subjects (with the material I consider important indicated in the preface and the .tex file available upon request). Beyond covering the basic material, I would highly advise spending considerable time simply doing problems from past prelims; it's good practice and familiarizes you with the topics that are usually tested. If for some reason you haven't found it yet, please check out the prelim Google doc for more resources generated by past students.
The Qualifying Exam for the Group in Logic & the Methodology of Science
The group in logic has three distinct options for the qualifying exam: mathematics, philosophy, and special. Each has different requirements which you can review here. In addition to the material on the group website, several forms related to the exam and a handout with a brief overview of the process can be found in the mathematics department office. I myself opted for the philosophy option, and so my thoughts and documents may be misleading for other selections. The philosophy option is intended to mimic the qualifying exam for the philosophy department with the only major divergence between the two exams being a diversity requirement in the philosophy department's. The qualifying examination itself is an oral examination in three subject areas with one faculty member from the group advising each. I would suggest initially approaching a faculty member or two with whom you are comfortable to discuss possible topics, mock up a very basic syllabus, and discuss which other faculty members might make good additions to the committee. Approaching the remainder of the committee is the next step; I've found it most effective to first email faculty with whom I had only a passing acquaintance, then either set up a meeting (if they respond) or drop by their office hours (if they don't respond). You'll likely find that different faculty have very different attitudes toward the exam. Some will be relatively hands-off and allow you to select large portions of the syllabus unilaterally; others will have much more concrete ideas about what topics and readings should be covered. If you have a strong preference in this regard, you'll need to ask older graduate students about their experiences with particular faculty. In general, polishing your syllabus and picking readings will be an ongoing process and conversation throughout your preparation for the exam.
After you've set your faculty members (three primary from the group itself and one member outside the group) and a general subject area for each of the three primary members, you should begin reading and meeting with them. It's common (and strongly suggested) to take notes on the papers/books as you read while also meeting regularly with your three primary advisors to discuss what you've read. Typically, preparation for the exam takes place during a single term in which you take no other courses. Two months before your anticipated exam date, you'll want to schedule the exam itself (a three-hour block during normal business hours), designate one of your primary advisors as the chair of the examination committee (cannot be your eventual advisor!), and fill out this form. It's expected (although not explicitly written in the rules) that you type up a critical overview (1800-3000 words) of each of your subject areas and send them to your committee members a week or so before the exam. For the exam itself, you'll need to reserve the room yourself (see the philosophy or mathematics department offices). Finally, the format of the exam (at least for the philosophy option) is simply, by topic, each advisor asking you questions within that area, possibly with side-questions from the other committee members. In my experience, it's unnecessary to memorize intricate proofs, though you should have a firm grasp of general techniques and any straightforward results. Axioms, assumptions, and important examples are fair game and will likely come up, as will the general contours of any philosophical arguments. If you can find other students with the necessary background, arranging a mock examination can be very beneficial in fine-tuning your presentation of key ideas and results (at the very least I would practice aloud answering the most obvious questions). After the oral examination itself, you'll be asked to leave the room while the committee members discuss; if all goes well, you'll be called back in and informed that you've passed. If not, there's a brief wait period before you'll need to go through the oral exam again. Good luck!