The Spartan II program pretty much requires special children who have specific gene markers. The original program started training the children at around age 6 and they were trained for years.
The Spartan III program was an attempt to speed up the process by being more lax on the gene requirements and the training time, and by the time of Earth's invasion hundreds had already gone through the program. The thing is that they weren't as "high quality" and were treated as kick-ass cannon fodder. If I recall correctly, there are about 10 left, all in the micro-dyson sphere.
From the way the program was conceived, I personally don't think they started other training groups. It was a secret project that involved kidnapping and faking the death of one of the Spartan IIs (Kurt) so that he could go off and train the new generation of Spartan IIIs along with CPO Mendez (who trained the original Spartan IIs). Kurt was killed near the end of the book. I got the impression that the Spartan III program described in the book was an act of desperation and the only one that existed because they wanted trainers involved with the Spartan II program to make the new one somewhat effective. But hey, they're all about secrets so... they could have new trainees somewhere.
(On a side note, Spartan II or III refer to the types of soldier rather than the sequential training of groups. For instance, there were several groups of Spartan IIIs who were trained at different times over the span of 15-20 years leading up to the invasion of Earth)
There is, however, one thing that has yet to be explained. In the first book there are three unaccounted Spartan IIs who have mysteriously disappeared and not mentioned again... In Ghosts of Onyx, the characters find 8 stasis tubes and bring them into the micro-dyson sphere. Five of those are Spartan IIIs who were placed in them by some events earlier in the book. The remaining three stasis tubes... unknown.. The book ends before the characters open them.