It seems the 5e rules are much more like the 3.Xe than the 4e. I also know a lot of players and DMs that simply refused to upgrade to the 4e rules, though they are now much more receptive to 5e, given some of the most advertised features. In this regard, what would be the "in a nutshell" summary of the game mechanics that have changed since 3.Xe? Would one need to go through 3.Xe → 4e and then 4e → 5e?
Note that this answer assumes knowledge and familarity with D&D 3.5e.
(Dis)advantage: A large number of situations and abilities where you would add a positive or negative modifier to a roll have been replaced with advantage (for positive modifiers) and disadvantage (for negative modifiers). This mechanic involves rolling twice and taking the higher result (for advantage) or the lowest result (for disadvantage). This simplifies a large number of abilities and mechanics, as well as reducing the ability for powergamers to abuse the system. (Advantage only works once, so they can't keep stacking it on a roll like modifiers.)
Proficiency: The majority of bonuses that scaled with your level have been replaced by proficiency. Essentially, anything you are 'proficient' in, you can add your 'proficiency bonus' to. So the bonus to hit in 3.5, that scaled with your level, is now your proficiency bonus, as long as you use a weapon you are proficient in. The extra attacks you gained with BAB are now a class feature of martial classes.
Skills: Assuming in 3.5 that you picked a few skills and kept them maxed, which is effectively a bonus to skill checks that scales with level, that is also replaced with your proficiency bonus on skills you are proficient in. (If you split your skill points all over the place, well, there's not really an equivalent.) Note that skill checks are now called ability checks; the only benefit of a skill is that it allows you to apply your proficiency bonus to relevant ability checks.
Saving throws: Most classes in 3.5 had one strong save progression and 2 weaker save progressions - the DCs have been adjusted, and the 'strong' saving throw is now the one you are proficient in and can apply your proficiency bonus to. Note that saving throws are now on each stat, but the stats matching the old saving throws (Con, Wis, Dex) are far more common than the new saving throws (Str, Int, Cha), at least in the material currently released.
Overall the proficiency mechanic means far less scores to keep track of on your character sheet, as well as being a primary source of 'bounded accuracy'. See this question for more on bounded accuracy.
Spellcasting, as mentioned in this answer, now scales depending on the level of the spell slot it is cast from rather than the player's level. Prepared spellcasters are also more flexible, preparing their 'spells known' for a day and then casting them in any combination that they have spell slots for. Cantrips have unlimited uses per day, so spellcasters always have something to do.
'Ritual' casting allows a spellcaster to cast without using a spell slot, but adds 10 minutes and can only work on certain spells. This is useful for spells like 'Augury' or 'Comprehend Languages' that you're unlikely to need often and will have plenty of time when you do need it.
Arcane spell failure is gone - if you're proficient with your armour, you can cast in it. If you're not, you can't.
Ability score increases as you level are now 2 points instead of 1, although still at every 4th level. Feats, however, are an optional replacement for an ability score increase. Because they require a fairly large sacrifice (2 ability points) to get, they are considerably more powerful than the majority of 3.5 feats (no more +1 to 1 type of roll feats).
Class features: All classes now have interesting and unique class features (rather than just more spells, or more feats). Each class chooses from a number of different paths (e.g. Evoker Wizard vs. Enchanter Wizard, or Champion Fighter vs. Battlemaster Fighter). This effectively replaces prestige classes. Multiclassing has gained ability requirements.
Racial features are also a bit more powerful - ability bonuses are no longer counteracted with ability penalties, and most races have unique and interesting features. Humans are still
Background: Your character now has a 'background'; in addition to their class, this adds proficiencies with skills and tools (and possibly even languages), as well as changing their starting equipment package (assuming you don't choose to buy starting equipment).
Character death involves a lot more luck. Negative hit points have been replaced with 'death saving throws' (DC10, no modifiers) that you make each round while you are on 0 hit points. If you fail 3 times, you die. If you succeed 3 times, you stabilize. A 20 stabilizes automatically, while a 1 counts as 2 failures.
Resting: Characters can take 'short rests' to quickly restore some HP. You have a number of 'Hit Dice' based on your class's hit die. So a 3rd level wizard has 3 Hit Dice, each of which is a d6. In a short rest, he can roll some or all of them and regain the total + Con. However, once they have been used in this way, it takes a 'Long Rest' (the standard 8 hours once per day), before they can be used again. A Long Rest also restores all of the character's HP, spell slots, and most resources used for class features.
Ranged and finesse weapons now allow you to add Dex to damage as well as hit. Finesse weapons don't require a feat to be used with Dex - it's up to you. 'Spell attacks' (replacing melee and ranged touch attacks) now use your spellcasting stat for the hit bonus.
Critical hits now let you roll all damage dice (base damage and, e.g. Sneak Attack) twice then add modifiers, rather than rolling base damage, adding static modifiers, doubling, then adding any additional damage dice. (e.g. Sneak Attack).
Roleplaying: There is a lot of discussion about 5e's focus on roleplaying. I'm not going to say anything about whether it is more or less focused on roleplaying than previous versions. The rules include guidelines (note, guidelines not rules) on creating a well-rounded character and roleplaying them, as well as an explicit mechanic for the DM to reward good roleplaying. The guidelines in question say that you can/should give your character a Bond (something their character cares deeply about), a Flaw in your character's personality, and an Ideal that your character strives for. The mechanic in question is Inspiration; the DM gives advantage on rolls to reward the player for roleplaying their character well, particularly for making choices that are sub-optimal from the player's perspective but match the choice the character would make.
This is a quick summary of the major changes, but you really should read the Basic PDF. It's short, sweet, and free.
There's A Grognard's Guide to 5E D&D Rules on Giants In The Playground forums, that not only summarises differences between the versions, but also gives a brief overview of the 5th edition rules.
As of writing this, it only has information from 5e Basic, and not the PHB.
D&D 5th edition did take some of the good parts of the hated 4th edition (and late 3.X edition) but pulled back from many of the 4e ideas that flopped. You have a limited ability to recover health between battles (and certain other resouces) through a "short rest" of 1 hour, thus extending the adventuring day.
Spells are quite potent at lower levels, but don't scale as automatically, i.e. a burning hands spell does 3d6 damage from 1st level or 20th level when used in a 1st level spell slot (you can use higher level spell slots to boost the spell, however). You will also notice that there are far fewer spells per day for casters, especially at higher levels, but cantrips (which have been buffed to be comparable to weapon attacks) are free to cast and certain spells can be cast as "rituals", taking 10 minutes more for the benefit of not taking a spell slot. There is a degree of free-casting for the wizard now: you can prepare a number of spells (besides cantrips) equal to level+int bonus and cast spontaneously from them, reducing your options each round but keeping your versatility (the system seems pretty good in this way).
Non-spell-casters seem to be better balanced with the spell-casters without making them feel too similar like was the case in 4e. There was a greater focus on role-playing than in 4e with the addition of backgrounds that have some mechanical effects.
One potential downside is the streamlining removed some detail. Skills are severely simplified. I haven't played enough to pass judgement completely, but it is definitely a leap in the right direction from 4E. Your wizard FEELS like a wizard. You might want to consider trying it out.
No, there is no official guide to what has changed. But this site and numerous other sites and blogs have compiled lists of changes, and many early "5E reviews" you can google will be heavy on listing changes to previous editions.
No, one would not need to go through 3.Xe → 4e and then 4e → 5e. There is no linear evolution leading from 3rd to 4th to 5th edition. But there are two possibilities to describe 5th edition if you want to reference earlier editions: