My DH and I provide fertility counseling for folks who are coming off of the Pill. We teach NFP, but I wanted to respond because there are similarities. We have been using NFP to postpone pregnancy for 3 years.
Because some hormones remain in the body for a longer period of time, and because each woman's body metabolizes these hormones differently, research indicates it may take between 3 and 6 months for your cycle to return to "normal". Most Pills have tiny print on them that says something similar. After about 3 months you will have a better understanding of what your fertility signs are like, and hopefully, your confidence of the method will grow, too. : )
That being said, you can definitely learn the FAM method while on the pill, or while in ransition. Just like Queen of Cups said. You can get used to noticing your body's natural fertility signs and charting your temp while still on the Pill, through your cervical fluid may be ambiguous. And the beauty of FAM and NFP is that your fertility signs will let you know what's going on, whether that is over a 28, a 32, or a 36 day cycle. Once you start charting, you may be surprised that your cycles are more regular than you think. Most cycle variation is caused by the pre-ovulation phase in your cycle (that is where stress, nutrition, and sleep come into play).
If you intend to stop taking the Pill, and use a natural method, it is advised that you abstain during the Phases I and II for about 3 months. That is with NFP, because we do not advocate contraception of any kind. If you are using FAM, then you would continue to contracept during Phase I and Phase II. Phase III is the post-ovulation infertility, so there would be no chance of pregnancy. Lastly, you asked about taking your temp? I wake up at 5:30 to go to the gym 3 days a week, and the other days I may wake at 7, 8, or 6. So, I take my temp at 5:30, and then fall back asleep if I am not getting up. This provides me with a temperature baseline. As for middle of the night waking, it may depend on the length of awake times and the frequency. Of course, if every night follows a similar pattern, as in breastfeeding, then that will become part of your "normal pattern".