"What is the best way to practice?" sounds like an easy question, but it's not. The answer I'm going to give is probably not what you want to hear, "Whatever works for you." Any method that works for you is going to keep you motived to keep practicing more and that is what leads to improved performance. Another way of putting it would be, "The best way to practice is the way that you enjoy the most" because that's what you're going to keep coming back to with your own interest and motivation. However, I'm going to focus on four aspects of practicing that can help clarify what steps can hold you accountable to doing your own practice: 1) Remind, 2) Record, 3) Review, 4) Report.
Everyone can benefit from simple reminders to do an action. A simple post-it note in a place that you can't avoid seeing it is a great start. Writing reminders on a calendar is simple enough, both in paper form and electronic form. There are many choices for reminder apps and I started doing research to find one that would be simple but also flexible for me to use on my own and also if I wanted to share my reminders with other people. That led me to Wunderlist because I liked the variety of formats and devices that it worked on and that it had more options than just an alarm-like reminder.
The phone version of Google calendar has the option of "set a goal" and the questions that it asks you are good ones for you to consider no matter what format of reminder you choose.
1) How often do you want to practice?
Once a week
3 times a week
5 times a week
2) How long do you want to practice for (each time)?
For practicing speech, it doesn't have to be a long period of time all at once. A short period of time such as 15 minutes or even 5 minutes is fine, it's consistency and awareness that are important. If you can focus your attention on your speech for just 15 minutes, or even focusing on someone else's speech by listening/watching a video and observing their speech, that will stay with you long after you finish your practice time.
3) When do you want to practice?
Whenever there's time on my calendar
In this meaning of "record" I am referring to "documentation" not audio recordings. You are recording (documenting) that you did it. Anything will work. I really like making to-do lists and checking items off as I do them. This could just be a checkmark or a note on your calendar to confirm that you did the practice. Bullet journals have become very popular lately and there are multiple formats and instructions and downloadable forms are easily found online. There's nothing incredibly new and different about these journals but the helpful concept is the same, create a goal, do it, record your action.
After you've created and set goals (practice sessions), set your reminders, performed the action, then recorded that you did them you will have a nice amount of evidence of the work you intended to do vs the work you actually did. At consistent intervals, take a look back and review your progress. Did you practice when you said you were going to? Do you have all those days on your calendar crossed off as "done"? Or is your list still sitting there without any checkmarks waiting for you to come practice? This is a factual reminder of why you may or may not be making progress with making changes in your speech. It can be a motivation for you to want to keep up a good record of practice or reset and try to achieve them next week.
Whatever format is going to be easiest for you to use and stick to is what's important: paper notebook, text notes on your phone, voice notes on your phone. You are reporting how you're doing and what you're observing other people doing. Examples include writing down what you overhear other people saying at work when they answer the phone that is different than what you say, and expressions that you hear but aren't sure how to use in your own conversation, or something that you just can't figure out what it is based on the sounds alone. These are notes that you can keep as reminders for yourself to try to figure out or bring to me and we'll work on it together. By observing what other people are saying and how they say it, you're increasing your awareness of the actual spoken English around you and how different it sounds compared to what it looks like. This should also give you more confidence to try something that might sound unusual to you and make you feel a little hesitant about saying it that way - if you hear everyone else sounding like that, you can do it, too.
Report on your own secret experiments of speaking in a different way.
1) Start with your expectations of what you expect to happen when you try speaking in a different way. How will the listeners react? How will you feel?
2) Try it. While you're doing it, in that speaking situation, you also have to be aware and observe the actual reactions of the listeners and your own feelings about it.
3) Report on how it went. Was it a positive experience? Keep in mind that a positive result is often hard to notice because you just got the correct food that you ordered, the listener understood your question, the audience laughed at your joke in your presentation, there's no negative reaction to report.
If it was a negative reaction (you got the wrong food, the listener asked you to repeat, the audience didn't know you were joking), then that's an important point to report on. Now, you can analyze what might have affected the situation and what you can try differently next time. Keep in mind that it may having less to do with your accent and more to do with the environment and external variables out of your control.
Your reporting can be shared with me if you have questions or would like advice, or it can be completely private and for your own observations of your progress. Consistent record keeping and reporting will provide you with a timeline that is more objective than your own memory of the progress you've made. Things that were confusing or intimidating three months ago might be familiar and part of your habit today. When we become comfortable, we stop noticing. Don't become comfortable in your bad habits, strive to become comfortable in your good habits.
Changing speech habits takes dedication to consistent practice, awareness, observation, reporting, and reviewing your progress. That shouldn't be intimidating though, it's the same process for any skill and you've mastered many skills in your life, some more difficult than others, and this one is no different -- the first step is difficult but then they get easier and easier. These tips are rather general, so if you have any specific tips, methods, or apps that have worked for you and you'd recommend, please let me know.