Ideally, good initial training will focus first, but by no means only, on stall and spin avoidance. With good pilot skill (though still in the training phase) stall recovery should be instinctive. A stall which threatens flight safety is going to happen when you're trying to do/not do something else in flight, and you're distracted. Recovery must be instinctive. Also, remind yourself, that it is perfectly acceptable for a well controlled stall to go unrecovered on every flight - the flare and touchdown.
It is possible in any light type (certainly some more easily than others) to aggravate a simple stall into something which turns into a spin. This is why you'll be taught to recognize and recover a stall promptly, but effectively, the first time. A buggered up stall recovery is taking you much closer to a spin.
The spin in a certified light aircraft, in and of itself, is not a serious problem, as long as you are going to correctly recover it, with lots of space around you, and without exceeding limitations. Do not go spinning without proper training and supervision, but certainly do go spinning. I am not an expert on spinning, as it is a very complex subject. But, I have spun about a dozen aircraft types during certification flight testing, and have experienced very different characterisitcs, and recovery conditions.
(Nearly) all ('cause I know nothing about Cirrus) certified aircraft are recoverable from a spin, it's a design requirement. Those which are not spin approved are so either because the manufacturer did not wish to have spin approval, or more commonly, because though recoverable, you get very close to limitations doing it. Add to that, that the recovery technique may be variable, based on configuration, and the techniques or cautions too variable to try to describe to the pilot in the flight manual.
My recent testing experience (on test flight permits) includes many stalls in two Piper Navajos, where, while trying to accomplish minimum altitude loss, it was easy to get a secondary stall, and have a rather messy secondary recovery. Certainly a spin could have been obtained doing that, if one was careless. In January, I did more than a dozen spins in a modified Cessna Grand Caravan. Though the handling of this larger aircraft in the spin is magnificent, it would be easy to exceed limitations during the recovery, without great caution. I installed a G meter, and was happy I did, as the recovery for a forward C of G gross weight spin required a 2.8G pull, at Vne. There's not much room to get that recovery wrong! The aft C of G handling and recovery, though also fine, was very different.
There's a video clip here: