To find out what you want to do after the MSc Logic, and then to succeed in getting what you want, you will have to talk to lots of people and look around for information in lots of places.
One of the roles of your mentor is to provide career advice to you (if you ask for it!). The same holds, maybe to a slightly lesser extent, for other members of the academic staff, particularly your thesis supervisor.
For instance, they might have suggestions for where to apply and they might even be willing to give you some feedback on your letter of application. Of course, they will typically be better informed about PhD opportunities than about other types of jobs, but this does not mean that it won't be useful to ask them about those as well (for instance, they might know a former student at the company you have your eyes on).
If you ask them for a letter of reference, make sure you ask a few weeks before the deadline and provide them with all the information they need to write a good letter for you (i.e., a copy of the job ad you are responding to and a draft of your own application materials, particularly your CV).
Talk to other students in the programme. The MoL Room is an outstanding source of expertise: many of your fellow students will already have done a significant amount of work on finding out how to get your dream PhD position, and many of them will succeed in doing so very soon. The MoL Room certainly is the best place to find out about impending deadlines, about how to ace a standardised graduate admissions test, or how to make it through the assessment centres of the big consulting firms.
You might also find it useful to talk to some of the PhD candidates at the ILLC (after all, they have all recently succeeded in getting a rather nice PhD position). Also look out for the Life after ILLC event organised by the PhD Council (usually around October), which is a good place to find out about non-academic career opportunities.
The UvA Student Careers Centre offers a range of services, such as providing feedback on your CV and application letters.
The ILLC website regularly carries announcements of relevant PhD positions.
Find out what the relevant mailing lists are for your main area(s) of interest, subscribe to them, and make it a habit of reading job announcements already before your time is up, so as to acquire some insights about how often you can expect what kind of openings at what places. This is a process that takes time, so start early.
Some of the most relevant lists include the LOGIC List and PHILOS-L. The best source for job announcements in linguistics is the Linguist List, which in fact is much more than just a mailing list. For AI, have a look at the Agents and Description Logics lists. For theoretical computer science, there is no standard mailing list, but see here for a few suggestions.