Even if you decide not to challenge the legality of your firing, you will be in a much better position to enforce all of your workplace rights if you carefully document the circumstances. For example, if you apply for unemployment insurance benefits and your former employer challenges that application, you will typically need to prove that you were dismissed for reasons that were not related to your misconduct.
There are a number of ways to document what happened. The easiest is to keep an employment diary where you record and date each significant work-related event such as performance reviews, commendations or reprimands, salary increases or decreases and even informal comments your supervisor makes to you about your work. Note the date, time and location for each event, which members of management were involved and whether or not witnesses were present.
Whenever possible, back up your log with materials issued by your employer, such as copies of the employee handbook, memos, brochures, employee orientation videos, and any written evaluations, commendations or criticisms of your work. However, don't take or copy any documents that your employer considers confidential -- this will come back to haunt you if you decide to file a lawsuit.
If a problem develops, ask to see your personnel file. Make a copy of all reports and reviews in it. And make a list of every single document the file contains. That way, if your employer later adds anything you will have proof that it was created after the fact.