From D&D 3.5 to 5.0(5E) - 26 - Aid Another to Give Your Buddy an Attack Advantage

If you feel like your character can't roll high enough to hit this monster, but maybe the Paladin standing next to you can, then you can forgo your attack action in D&D 5.0 and instead choose to use the "Help" action to help the Paladin hit the monster on her attack.  Simply tell the DM that you are going to put your thumbs in your ears, wiggle your fingers in the direction of the monster while doing a little dance.  If the Paladin then choose to attack the monster before your next turn, the Paladin will have advantage on her first attack.

From D&D 3.5 to 5.0(5E) - 25 - Non-Lethal Damage in 5th Edition. A Knock Out!

You won't find non-lethal damage in 5th edition, but when it comes time to strike the killing blow the attacker can show some restraint and "knock out" their opponent instead.  This only applies to melee attacks.  But if you character is one of those that wants to show mercy, or you want to tie the monster up before waking it up to interrogate it, then a "knock out" is for you.  Simply tell the DM, at the time you deliver the blow that would have put the creature at zero hit points, that you would like that damage to knock them out and leave them stable.  As the book says, the attacker can make this decision the instant the damage is dealt.

From D&D 3.5 to 5.0(5E) - 24 - Standing Up from Prone

It is a lot easier to stand up in D&D 5.0 than it was in 3.5.  Hey, everything is easier!  This change is simple.  Instead of requiring a "move" action to stand up as we did in 3.5, you consume one half of your move action distance while standing up, thus giving you the ability to stand up and move within the same turn.

If your movement is 30', you can stand up, then move an additional 15'.

If your movement is 40', you can stand up then move an additional 20'.  Apparently it is harder for Elven monks to stand up (taking 20' of their movement) than it is for Dwarven clerics who can stand up using 15' of their 25' movement and can then still move 10'.

If you want to move before standing then you are crawling and for every 1' you crawl you use 2' of your movement.  So if you want to cover much ground it is almost always better to stand up first.

From D&D 3.5 to 5.0(5E) - 23 - Initiative Order Doesn't Change

Shuffling character initiative orders during combat is another victim of the rules simplifications in 5.0.  I am not sad to see it go, although it can allow a character to have back to back turns.

For example, if my fighter is in a narrow hallway behind my friend the monk, and the monk is fighting an orc, the combat could go like this:

Round 1:

1. Fighter "readies an action" to attack the orc if the orc drops the monk and steps forward.
2. Monk attacks the orc and hits two times for 10HP.
3. Orc attacks the monk and hits for 8HP knocking the monk unconscious. The orc steps into the monk's square.
4. Fighter uses his "readied action" to attack the orc and hits for 6HP.

Round 2:

1. Fighter uses his action to attack the orc and hits for 9HP.
2. Monk, rolls a death saving throw.
3. Orc attacks the Fighter and misses

In the above example, the fighter got to take two actions in a row.  That will happen occasionally, but most the time it will probably not occur.

The thing to remember for 5.0 is to leave the initiative order as it was in every round throughout the battle.

From D&D 3.5 to 5.0(5E) - 22 - Where Did My 5' Step Go? Disengage

It is gone.  D&D 5.0 doesn't have the idea of a 5' step.  But you don't need it for movement since you can always move 5', then do something like take an attack, then move some more.  Since 'attacks of opportunity', now known as 'opportunity attacks' still exist, you might want to consider finishing the first monster you attack before moving toward another foe.

You also can't use a 5' step to move out of an attacker's range for free.  If you want to stop fighting the enemy you are standing next to you need to use the "Disengage" action to prevent your enemy from making an opportunity attack.  In fact, while you are "Disengage" you can move past several enemies avoiding opportunity attacks from each of them.  The distance you can move on your turn during "Disengage" is the same as your movement rate.  So if you can move 30' on your turn, you can move a total of 30' on your turn while disengaging.  You can't disengage for 30' then move an additional 30'.

From D&D 3.5 to 5.0(5E) - 21 - Recovering Hit Dice - Different Than Hit Points

By now you probably know that when your character has a long rest they recover ALL of their hit points, even if you only had 1 at the end of the day and your hit point maximum is 100.  You might feel like Loki after confronting the Hulk at the end of the day, but your will feel 100% better the next morning.
You may also know that during the day you may spend some of your "Hit Dice" after a "short rest" to regain hit points.  This is a new use for the concept of "Hit Dice" that was not present in 3.5.
If my fighter is 8th level, then he has 8 Hit Dice and is an 8d10 creature.  Let's pretend the fighter has a 72HP maximum.  During the morning you fight off some small dragons and after the battle you are still down 35HP of damage.  You are on your own with no healing.  You can take a short rest, and then expend some of your "Hit Dice" to recover some hit points.

1. You decide to expend 3 of your "Hit Dice" to regain hit points so your roll a d10 three times.  (Fighters roll a d10 because they use d10 "Hit Dice".  If your character is a monk he would roll a d8 instead of a d10.  Check your class description for the Hit Dice roll to use.
2. As you roll the 3d10 you get a 7, a 4, and a 2 for a total of 13HP.
3. You are now down 22HP instead of 35HP.
4. But you would like to recover more hit points, so you tell the DM that you are going to spend 2 more of your 8 Hit Dice.
5. You roll a 7 and an 8 for a total of 15HP.
6. You are now down 7HP instead of 22HP.  Much better.
7. You have spent 5 of your 8 Hit Dice for the day.

The rest of your day is uneventful; you take your long rest and recover all of the hit points you were down.  You are now at your maximum of 72HP again.  But you have NOT recovered all of your Hit Dice.  You are starting out the new day with only 7 of you 8 Hit Dice available to you.  This is because the rules state, on page 171 of the PHB, that you recover just "half" of your Hit Dice each day at a maximum.  Since your maximum Hit Dice is 8, you can only recover 4 of your Hit Dice after each long rest.

I think this is to discourage players from using your Hit Dice too casually.  You want to put some thought into it each day if you expect the next day may also be challenging.

From D&D 3.5 to 5.0(5E) - 20 - Dash Instead of Move/Move

Move/Move and Run are gone from the combat mechanics in 5.0.  As with most of the rule changes, the goal is to simplify the mechanics even at the cost of being less realistic in order to speed the game play.
Characters never, or perhaps very rarely, need to choose between a full round action, a standard action, and a move action, because in 5.0 you pretty much always get both:

• An Action
• A Move

If you want to move more than your move allows, then you are employing a "Dash".  So "Move/Move" in 3.5 basically equals "Dash" in 5.0.  This does not allow you to cover three times your movement distance on your turn.  If you choose to Dash action for your action, you gain the ability to move your movement distance.  So, if your movement is 30', the total distance you can move in a round is 60' (30' from your "move" and 30' more feat from choosing the "Dash" action).

There is no "Run", so your maximum movement is basically your movement times 2.

Don't forget, you can break up your move in 5th Edition mechanics.  So you can move 10', take your action, move another 10', take your bonus action (depending upon the bonus action), then move another 10' if your movement rate is 30' or better.