If you have the spare money, train your employees first. That will get you good stat bonuses for cheaper in the long run. Level them up once they stop gaining stats from all available training methods.
Leveling up staff doesn't net you significant stat bonuses for most job titles. Producers, Directors, and Hackers see a larger stat jump from leveling up than the others, but overall the stat gain is not as much as you will get from training the staff person over and over until they can no longer gain additional stats. In addition, leveling up the staff member has a reoccurring cost in terms of salary. For job titles that are more advanced, the difference in their first year's boosted salary will be more than it would cost to train them fully.
The strength of leveling your employees is that you can use a career change manual to change their job title once they hit level 5. You can do this over and over again to max out their stats and create a character that is level 5 in all job titles (Coder, Writer, Designer, Sound Engineer, Director, Producer, Hardware Engineer, and Hacker). However, this comes at the cost of a huge salary.
As a point of reference, I did this with Francoise Bloom.
- Final Title: Hacker
- Salary: $44157.6K
- Program: 825
- Scenario: 817
- Graphics: 534
- Sound: 471
As you can see, the trade-off for those ridiculous stats is a huge salary. Overall, you can achieve "Game of the Year" status by simply hiring all of the best people, training them until they cannot train any more, and at most leveling them to level 5 in their current role without changing their title. I did this in my first play-through and had several "Game of the Year" titles, and overall the training contributed more stats to each person than the leveling had done.
To address an implied misconception in your question: you can train employees an unlimited amount of times. However, after a certain number of times, they stop gaining stats when you use that training method and will start gaining 0 in all stats. You can still get stats by getting lucky enough to get a "super parameter" result even after a standard result nets you 0 additional points. At this point though, farming for the "super parameter" result will cost you a lot of money for low return.
Also, keep in mind that you will have access to new training methods every time you move into a new office. If you have maxxed out usage of existing methods and you move into a new office, you should begin training your employees with the new methods until they max out.