To clarify even further, there are certain ASELs for which the FAA (as opposed to an FBO or insurance company) will require additional training and endorsements. Complex, high performance, and tailwheel airplanes are three of them.

But, of course, the privileges and limitations of your new pilot certificate were covered in your training and will probably be asked about in the oral part of the checkride - after all, it's the very first PTS task.

In terms of insurance/FBO requirements: Generally, all it involves is a "checkout." Some study, perhaps a written quiz that covers the operating characteristics or the airplane, and a flight with a CFI to run through a series of tasks to show that you can handle this make/model. The first checkout is generally the most difficult (relatively speaking). It's sort of like the first time you drove a different car after you got your drivers license - it probably felt a little weird not having the same references you were used to in your "training car." But like the car, the aircraft transitions get easier as you do more of them and realize the the similarities of these airplanes far outweigh their differences.

So get that private certificate and have some fun!