Well let me preface this by saying I never actualy played 1st edition. and, that I do love DnD although I dislike certain DNDisms. I realize that DnD started out as Gygax and Arensons personal game that was done how they wanted it. However, it has become more than that...it has become a generic fantasy game in terms of how its used...the design just doesnt always reflect that,
So here goes:
Fire and forget magic as the only option. the Vancian stories are the only ones I know of that have magic working this way. Related to this, the way that magic and the classes are dealt with, in several respects. Wizards supposedly gain ALL there power from study and learning. No innate gift, no nothing. Its all simply because they learn how to do it. I cant recall a story offhand wherein magic was like that...no innate spark no nothing like that. This also leads to the whole spellbook thing...a wizards power is tied entirly to there spellbooks.
the Arcane/divine magic divide, and clerics. Again this is something not seen in fantasy and mythology...there are often different forms and discplines of magic, but rarely are they as exclusive as arcane/divine is in DnD. In most things magic is a ohesive whole, even if there are different discplines or even forms of it. The division of what each kind can do in DnD is also rather steep. Arcane magic isnt allowed to heal(aside from bards for some reason), it purely the domain of the divine. same with spells dealing directly with good and evil. I think magic should just be magic.
the cleric class, first has balance issues and second(and linked to the arcane/divine divide) is a role that I dont really feel fits into high fantasy. Really I dont see the need for a priest CLASS at all since anyone can really fill the role. However, I can see having one perhaps, although having the priest class also automaticaly be a strong combat class to me seems odd, and also makes the Paladin class redundant. And since the Paladin, in my mind, fits the things that the cleric is based on(templars etc) better, its the cleric that needs fixing.
There shouldnt be one healing class. generaly, anyone who can use magic in literature can use it to heal, at least somewhat.
The dependency on magic items is unpleasant on a great many levels.
Short elves are annoying.
The alignment system is annoying. I dont having one, and having mechanics with it, but as is said before a more definite stand should be taken on it. I also find the Law/Chaos part of it entirly absurd.
I dont mind subraces. And yes other than humans there are at least 2 or 3 of most of the common races. Some are somewhat redundant however.
Linked to my whole arcane/divine magic thing, I find many spells to be drasticaly misplaced. the fact that wizards dont get animate objects, spell immunity, spell resistance, find the path, air walk, and various other spells that just seem like basic magical effects to me, is very strange.
Alignment restrictions, and especialy the paladin's and the fact that a Paladin must be Sir Galahad or loose his or her powers.
The fact that no matter how much you practice you cant fight with any 2 weapons without penalty.
I do like the current bard but I agree that the fop image is overdone and it should be easier to have a bard who's truly an imposing figure
To an extent, the whole hp/ac/abstract combat thing. I dont see that as being core to the nature of DnD, I think a better system could be designed and used, and only improve the flavour.
I think that pretty much covers the biggies

course theres lots of little details here and there but most are unimportant.