It's Complicated...
I use a very non-traditional and reality-based notion of training... It's complicated!
I Joey Balboa, Rogue Extraordinaire, who (at first level) concentrated upon Hide, Move Silently, Listen, Search, Spot, Disable Dvice, Open Locks, and Sleight of Hand suddenly wants to become a Fighter, then he has some explaining to do... How did he pick up that Feat (and what is it)? He had proficiency in a few weapons and light armor, but how did he suddenly learn all the rest, and Medium & Heavy armor, plus buckler and three types of shields?
Now if he carried an Exotic Weapon and shield, and wore non-light armor through first level, no problem. He learned by doing. Once he has the XP, he can advance to Rogue/Fighter.
If not, then the easiest thing to do is to go to the equivalent of a Dojo, and be trained by a bunch of Fighters in weapons, armor, and shields. He'll be charged, and take some time, but no biggie...
If Joe Fighter wants to learn lock picking, then he needs to explain how, too... He can buy a lock and some picks, and learn on his own, or work with a local locksmith, or go down to the local Thieves' Guild (if there is one), enroll, become an "Associate Member", pay the fee, and take Lockpicking 101.
Now if Joey Balboa Fighter/Rogue wants to pick up some skill in Diplomacy, as a Rogue, then no problem. He has been trying to convince people for almost three levels, and needs no equipment or training.
On the other hand, when he first became a Fighter, and wanted to learn how to ride, he would have needed a horse, saddle, and bridle. He might own one, or borrow one, but he could also hire the local stable boy for a few coppers...
A PC who already knows a skill, or who is trying to use one they don't have, can advance it with no problems, usually. Learning locksmithing requires some stuff, but riding bareback could be learned without reins and saddle. Advancing skills you already have is no problem, unless you've lost your equipment. Even then, it's probably possible.
So, I had a PC wanting to be a Ranger, who began play as a citified Rogue... He took all the Search, Spot, Listen, Hide, and Move Silently that he could, and a little Swim, Climb, Rope Use, etc. When he finally became second level, he learned a bit of the other skills through experience, learned to ride the horse he'd been riding through first level, started learning to train it, and then sank the rest of the points into Wilderness Lore/Survival, which he was learning firsthand.
After that, he had no "training" issues until he wanted to know spells, and learn some Concentration. The only things really needing explanation were his use of shields and Martial Weapons. Since he bought a buckler at second level, and never used non-rogue weapons, this was just assumed to have been learned in the months between adventures...
My DMing style is to provide the PCs with floods of Versimilitude, and my "training" requirements go right along with this. Where and how you learned something is important, and you can then improve upon this pre-existing knowledge (or not) at your discretion...