First thank you for these comprehensive responses, they are so far more than I expected.
I will respond first to the questions that 5ekyu posed as they are more in depth than I had even really thought of myself, and I think they cover what everyone else has asked. My general dislike for the 5e system stems from my feeling that it is too limited in scope, that the skills are so broad that I'm often being left with uncertainty about how to apply them. Also, I have found that more comprehensive systems give players ideas on how to use skills themselves they may not have thought of.
Can you name two things that the 5e skills system *currently does* that you like and want to keep?
I do like that the current system is simple, and it meshes well with the manner in which D&D is played in that it uses a D20 and is tied to both the proficency bonus and the attribute bonus. I like this and would like to keep it.
Can you name two things that the 5e skills system *currently does* that you dont like and want to scrap?
As I said above, I'd like a system that offers more skills. I really love the Shadowrun system and how skills work there, which is entirely inappropriate for D&D I understand, however the depth and scope of the skills offered lends itself to suggesting to players how to use them, and helps to define characters.
Can you name two things that the 5e skills system *currently does not do* that you like and want to add to it?
I'm certain that the current system can accommodate all that I would like a more robust system to handle, however as stated a couple of times now, I do not like how general it is.
Can you name two things that the 5e skills *currently does not do* that you don't like and want to make sure doesn't get added by your new changes?
My group is filled with new players. Though they have been gaming together now for nearly two years, they have only gamed together with very little variation in the player base. I'd like to expose them to a broader skills system that would lead them to think of using skills in new ways. I think the 5e system would allow them to do most anything they can think of currently, but I wish for them to see new role-playing opportunities that a more robust skill system would suggest to them.
Thank you for your time.
This. Read the Intro section of the PHB and the Basic rules for the core play loop of:It's also worth noting that, given the rules on How to Play and adjudicate actions, there is a fairly stark difference in game play between D&D 3.Xe and D&D 5e in this area. The former is much more mechanics forward, for lack of a better term, where players are expected to "use skills." In D&D 5e, players just describe what they want to do and the DM decides whether the proposed action is successful, unsuccessful, or if there's uncertainty as to the outcome. If there is uncertainty as to the outcome and a meaningful consequence of failure, the DM calls for an ability check. Otherwise he or she just says what happens.
You don't have to play that way, of course, but that is what the rules say to do and it may help inform what choices you ultimately make for your house rules and how that may impact the intended play experience.
It's also worth noting that, given the rules on How to Play and adjudicate actions, there is a fairly stark difference in game play between D&D 3.Xe and D&D 5e in this area. The former is much more mechanics forward, for lack of a better term, where players are expected to "use skills." In D&D 5e, players just describe what they want to do and the DM decides whether the proposed action is successful, unsuccessful, or if there's uncertainty as to the outcome. If there is uncertainty as to the outcome and a meaningful consequence of failure, the DM calls for an ability check. Otherwise he or she just says what happens.
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It's funny because this how I play every system that has skills, 3e included.
There has been some great advice here that I don't think I could improve on. I particularly like the idea of backgrounds giving general knowledge and background not tied to specific skills.
There are times when I ask my players if it's okay if we do a 1-shot with a different style or rule implementation just to see if it "fits". There is no one way that's going to work for everyone, so experiment a bit.
In addition to what others have said, I'd just say that I'm pretty flexible on how skills are applied. Want to intimidate or assist in intimidating someone using a strength (athletics) check to lift them up in the air one handed? Go for it! Might backfire depending on the NPC though, so I might ask for a wisdom (insight) check before you try it.
In my experience the "looser" I can make the implementation of the rules while abiding by the spirit of the rules while still being consistent the more creativity I seem to get out of my players. Don't get me wrong, I enjoy video games that have skill trees as an example, but I just don't think it's a great fit for this version of the game.
It's funny because this how I play every system that has skills, 3e included. "I want to do this, do I need to roll anything?" "Yes, skill x is most appropriate."
I think you are describing an rp style. Maybe 5e encourages that style more than others, I don't know.
And sometimes rolling a skill is appropriate even when success is guaranteed because:
a) my player has invested heavily in a skill and 'wants' to roll a dice and show off how cool his character is at doing skill (x). Who am I do deny a player a chance to throw a die if it brings them joy? Players tend to approach problems using the tools they are best at and, often, like to show those things off.
b) Sometimes I use degrees of success which doesn't usually affect anything other than narrative. So it lets a player or GM narrate a cooler description or outcome if they succeed with style.
Usually 'b' comes as a result of 'a'. I'm happy to narrate a cool success without the use of dice too.
[MENTION=23718]twofalls[/MENTION]
The skill list in 5e is good.
Expand the use of Investigation a bit and make Perception less of a catch-all
Use Medicine more
If you were looking for a way to incorporate point-buy skills in 5e, then I'd probably have suggestions but it doesn't look that way. [MENTION=6919838]5ekyu[/MENTION] had lots of good suggestions.
- As suggested, Definitely separate stats from skills. Use the skill/stat combinations that best suit the situation/action.
- I've done away with 'tools as skill proficiency' and just made each tool a 'profession' skill. A lack of tools either makes it impossible to do that skill or gives you disadvantage. It hasn't changed much - it's really mostly a change in semantics (using Burglary instead of 'Thieve's Tools', for instance) but I like the change.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.