Your employee handbook should have a policy that puts employees on notice that poor performance or bad behavior will lead to discipline, but it should not lay out a specific escalation process. If you tell employees that they'll be subject to a certain set of steps or entitled to a certain process, they can hold you to that policy later and claim they've been discriminated against if they don't receive the benefit of each step in the process.
And there are of course occasions in which progressive discipline should not be used. Violence or threats of violence, for example, often merit immediate termination following an expedient investigation.
Second, it's imperative that you document everything throughout the progressive discipline process.
Even a verbal warning
should be noted in the employee's personnel file. If something isn't in writing, it might as well not have happened.
Third, whenever possible (and practical) have another person in the room for these conversations. Having an HR representative or other manager present to witness disciplinary actions will reduce your risk if an employee later claims the conversation went down differently than it did.
Fourth, stick to facts. You're disciplining an employee because they've done something wrong or poorly. Focusing on the facts of the case provides the employee with a clear path going forward and protects you if the employee later challenges you on any of the steps of the process.
Finally, be compassionate, but not apologetic, particularly if you've reached the point of termination. If you've followed your internal progressive discipline policy and the employee has failed to improve, the employee shouldn't be surprised if termination occurs.