I have been writing my diaries since my first training in Shotokan karate on October 1st, 1991. In 1991-1999 these were the handwritten notes, since 1999 these are the electronic text files.
I have few different types of training diaries.
1. The detailed description of every training with comments on correct performance, important things, personal errors and ways of improvement.
The main advantage is that you can easily recall what you were doing some time (week, month, 10 years, etc.) ago. Of course, it takes some time to write everything down, but, while writing down, you can review the obtained experience and devise new training plans and strategies for yourself.
I have written down 99,9% of all training sessions I have ever did or taught. 0.1% falls out because I was too tired or too busy to write down. But never I was lazy to write!
2. Personal training plans: these are the plans for a week, month, year, etc.
If the previous one is "the past", then this one is "the future". I write down and constantly update my personal training goals, training schedules, etc. This is very helpful if you want to have your training in a systematic way.
I don't understand those people who simply come to train without analyzing why they do this or that. If you want to progress you have to set some goals. These goals can be global, like "I want to "improve my kata", or "I want to become nidan", or "I want to win this tournament". Having a global goal can be very inspiring, but in order to progress toward this goal you have to set smaller, local goals. Thus, "I want to improve my kata" leads to "I want to improve my Heian kata", then leading to "I want to improve my Heian Shodan", then leading to "I want to improve my gedan barai (or age uke or my balance in 270 degree turn, or ...)". A local goal is very explicit, specific and gives you a specific direction for a real improvement by regular training.
3. Separate files on specific topics.
These are the collections of information on those specific topics, which I am interested. Examples: notes on learning specific combination for kumite, detailed comments of my current tokui kata, etc.
If you would like to start writing your own training diary, then, as a first step, I would suggest to understand why would you want to do it. Writing a diary regularly requires either motivation or discipline, or both.
One of the possibilities to start is to simply write down a short (or detailed) description of the training session. But I would recommend to add your own notes to this description like important points, your errors, your successes.
Another possibility to start is to separately write down your errors and successes mentioned by your sensei during the recent class. But this list should not be a simple reflection of what has happened, like a historical record. Well, a historical record could be interesting to read later, but we are aiming on the progress, aren't we? Thus, this list of errors should become a basis for your self-training plan.
Also, as I suggested earlier, you could combine this list or errors/successes with a short (or detailed) description of the training session.
Speaking of training plans.. Writing down reasonable and realistic training plans requires some understanding of methodology, which comes with experience. But the steps for the correction of those errors, pointed out during the recent class, could be a good start. And of course you could ask your sensei to help you in devising such a training plan.
P.S. As always this post can be updated in the future. I like to keep the information in one place. So, if there will be something else to say on the topic "training diaries" I will simply add it to this post. The announcement about an updated post will be given at the top of the page.