You cannot learn spells of another class without actually multi-classing or picking up a feat like Ritual Caster or Magic Initiate.
While there is some overlap in class spell lists you can only ever learn the spells of your class unless you make the choice to multi-class or forgo the stat increase to pickup one of the two feats I noted above.
Proficiency or language training is allowed, though limited by DM Fiat.
The PHB notes:
First, you must find an instructor willing to teach you.
The DM determines how long it takes, and whether one
or more ability checks are required.
The training lasts for 250 days and costs 1 gp per day.
After you spend the requisite amount o f time and money,
you learn the new language or gain proficiency with
the new tool. - PHB Part 2 Adventuring p.187
So long as your DM feels the PC is a skilled enough instructor then the DM will arbitrate what else in necessary. The costs and length of time should still be the same, they can either be paid from one PC to the other or be used up as "training costs" for materials over the 250 days, again at DM discretion. As waxeagle noted, this would take up the downtime days of both the PC learning and the PC training, precluding them from other tasks.
You could apply this same rules to learning new skills.
PHB doesn't specifically mention anywhere about learning new skills as part of a down-time activity. However I think its perfectly reasonable as a GM to allow players to learn skills from one another in the way I outlined above.