Unfortunately, not all the employees you hire will perform at their best or even follow all of the rules. When you need to "discipline", you will need to do so wisely and proficiently.
Let's talk about it. So you have an employee who's consistently breaking the rules and/or "under performing", you're tired of their behavior and now you're ready to terminate them. What steps have you taken to improve the situation? Did you even try?
If you're not sure how to alleviate this "problem" smoothly, follow these steps:
Counseling: Does the employee know what's expected of them? Try meeting with the employee and going over the role and company expectations. Sometimes people can become complacent if they've been at the same company too long and not progressing, or they don't quite adjust well to new environments. Find out what the challenges might be.
Training: I would say retrain but more often than less, newly hired employees are not trained properly enough to perform in their role successfully. If performance is the issue, the employee could be lacking the necessary skills and/or knowledge to perform the essential job functions. Try setting up the employee with a few trainings that are related to improving job skills.
Verbal Warning: So the employee knows a) what's expected of them and b) has received proper training. Now what? It's warning time! Have a formal conversation with the employee. Confront and address the problem, state the expectations going forward, and most importantly - DOCUMENT! Keep a record of what was said and when it was said. This documentation is not a formal writeup, it is just documentation that the conversation occurred.
Written Warning: Recap on what the problem or issue is. Discuss the prior events and what actions will be taken if the events continued. This particular meeting has everything spelled out in a written documentation. The manager and/or Human Resources will sign off on this, along with the employee. The document explains the problem in detail, the measures that were taken to provide a solution, previous events, and the outcome if the event(s) were to occur again.
Decision-Making Leave: If the employee continues to have performance issues after the written warning, they should be placed on a "decision-making leave". Also can be called a suspension. This leave can be paid or unpaid - it's at the employer's discretion. During this leave, the employee has time to reflect on their actions. When they return to work they're either going to work on improving their behavior or resign from the position.
Termination: If and when you reach this phase, you should feel confident about termination. It is not you terminating the employee, the employee is terminating themselves. They were counseled, (re)trained, and provided with the tools necessary to improve on their performance. Your risk for a lawsuit on wrongful termination is minimal.
For more difficult employee situations, your organization should seek legal counsel.