Only If Sources Beyond the Player's Handbook Are Used
Omitted from the online sorcerer description is the following from the Player's Handbook:
[The sorcerer's] new spells can be common spells chosen from the sorcerer/wizard spell list (page 192), or they can be unusual spells that the sorcerer has gained some understanding of by study. For example, a sorcerer with a scroll or spellbook detailing an unusual sorcerer/wizard spell (one not on the sorcerer/wizard spell list in this book) could select that spell as one of his new spells for attaining a new level, provided that it is of the right spell level. (54)
Emphasis mine. Using a very hard reading of the rules as written, if the DM determines the sorcerer can gain "some understanding of [the spell cure light wounds [conj] (PH 215-6)] by study" from the scroll and the spell cure light wounds appears on the Sor/Wiz spell list in another source other than the PH, the sorcerer can add it to his spells known.
As the above is largely campaign-dependent, it needn't be a scroll. The DM determines what exactly the sorcerer can study to gain this understanding (e.g. ancient dragon scales possessing the secrets of weird magic, the corpses of magical minions, the drippings of sacred candles).
I am, however, unfamiliar with a published setting or source that adds cure spells specifically and directly to the Sor/Wiz spell list.
However, another source can totally be the DM's campaign notes. Thus, in a campaign that amends the Sor/Wiz spell list, the sorcerer could, upon understanding the spell via study, select that spell as a new spell when he reaches the next level if the spell's on the Sor/Wiz list at the appropriate level.
The DM should look askance at a player who claims his authorship of another source on a cocktail napkin amends his character's spell list, unless the source is accompanied by an appropriate bribe.
There are many ways to gain additional skill proficiencies in the rules.
If you or your players want to make a new set of proficiencies, you're free to do so using a customized background: (PHB 125)
You might want to tweak some of the features of a background so it better fits your character or the campaign setting. To customize a background, you can replace one feature with any other one, choose any two skills, and choose a total of two tool proficiencies or languages from the sample backgrounds.
The Skilled feat (PHB 170) is self-explanatory:
You gain proficiency in any combination of three skills or tools of your choice.
As a DM, you can let your players get training in skills in order to gain proficiency: (DMG 231)
A character might be offered special training in lieu of a financial reward. This kind of training isn't widely available and thus is highly desirable ... Possible training benefits include ...The character gains proficiency in a skill.
However, consider that this bonus is in the same list as gaining an extra feat, so it's not something to be done lightly.
Finally, both the rogue and the bard have ways of boosting their skills. Rogues can double their proficiency bonus for a handful of skills through the Expertise feature (PHB 96) and bards get half of their proficiency bonus to any check that doesn't include their proficiency bonus (PHB 54).
You might be interested in the Background Proficiency variant rule
DMG 264 has the "Background Proficiency" variant rule, which seems to hew pretty closely to the mechanics you desire:
With this variant rule, characters don't have skill or tool proficiencies. Anything that would grant the character a skill or tool proficiency provides no benefit. Instead, a character can add his or her proficiency bonus to any ability check to which the character's prior training and experience (reflected in the character's background) reasonably applies. The DM is the ultimate judge of whether the character's background applies.
You don't need specific rules to make crazy situations
Why did the designers get rid of the skill point system in the first place? Mike Mearls said:
We unified the progression for skills and weapons under one set of rules. That removed a lot of complexity and allowed us to include skills in the simplest version of the game without adding a lot of rules overhead.
Therefore, instead of depending on an arguably unwieldy and complicated set of rules for skills, you can use the simplified framework of 5e to create the kinds of specific "proficiencies" that you describe in your question.
More specifically, you can let your players add their proficiency bonus to specific kinds of checks, or even give advantage for more significant bonuses. If your character is an expert in astrology, maybe let them add their proficiency bonus to Arcana, Nature, or Religion checks that involve the stars. Examples of this kind of bonus exist throughout the system: for instance, the Belt of Dwarvenkind gives advantage on persuasion checks against dwarves.
Indeed, such cases are what the advantage mechanic was made for: the DM giving quick bonuses in response to favorable situations. You might find that this goes a lot smoother than a skill point based system. As a player, I've found that DM bonuses like this are not only balanced (because they're highly situational), but also a lot easier to keep track of than, say, 3.5's super complicated skill system.
Best Answer
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:
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.