I spoke a bit about gaining experience in a previous question, but would also add this based on your question. My first job for a race team was essentially to sweep the floors. I helped out wherever an extra person was needed, sometimes it was in the bodyshop, other times putting together suspension, etc. Over time, the jobs I helped on got more sophisticated and at some point I started going to tests and races.
The team was Downing Racing, which designed, built and developed its own (Kudzu) sports car chassis running Mazda rotary engines. As such, in some ways, it was a young engineer’s dream as the team was always designing and developing but that was not part of my job with the team. As much as I wanted to poke my nose around the engineering office, my job was to mechanic/fabricate/general lackey on the car. So I did the best I could at those jobs. It was a great experience and likely too many engineers do not have enough experience getting their hands dirty working on race cars. Similarly, when I got a job as a DAG with a Champ Car team, my personal interest did not lie in wiring looms, Lemo (now Autosport) connectors, electronics or networks, however that was the job I accepted and I did the best I could in that position in hopes that I would move to a position eventually that would better suit my engineering interests. Essentially if I did a poor job at my current position, what confidence would I give to the team I was working for to move me into any other position? Having said this, after hours once my work was completed, did I catch up on the engineering reports from the previous event or discuss topics that I wanted to learn more about with the more senior colleagues on the team? Absolutely, but I made sure that I completed the task that was my primary responsibility on the team.
As far as what series would be best for training a young engineer, I would say that they all have benefits and drawbacks, as will the teams in the series, and it will depend on your goals. To start with, does your interest lie in formula cars, sports cars, circle track or production cars? If your heart lies in NASCAR, working in a small formula car series is not likely the most efficient route. After this is determined, have a look at the series that are likely to be able to support a full-time position. Most regional racing series do not have the budget for full-time people let alone someone in an engineering capacity. Certainly if you cannot find a job otherwise, starting out volunteering for one of these teams might be a good option. Generally the bigger the team/engineering group, the more specific the position. So you are likely to learn a lot about whatever specific role you fill for the team. On smaller teams with less engineers, generally the role is more diverse so you will be exposed to a wider range of motorsports engineering topics. For example, a larger team might have a technical director or chief engineer, race engineer, design engineer, assistant or performance engineer, simulation engineer, DAG, development engineer (who might handle wind tunnel testing, rig testing, special projects, etc.) or some combination of this, whereas a smaller team might only have a race engineer and a DAG. While the smaller team will have to manage the workload among less people it will also likely do less testing (track, wind tunnel, rig or otherwise) as the budget is not available to support such programs.