I brought up some of these ideas on RPGnet, but I'll elaborate here. Granted, this will be gone in a couple of days, so I think I'll be saving it to continue the discussion on the new server.
The basic idea for streamlining D&D would be the KISS principle. In general, the PHB should be a toolkit to allow players and DMs to create characters appropriate to a variety of fantasy adventure types. The key words here are fantasy and adventure, not Europe, Tolkien, Howard, etc. While I think it's fine to be open about being inspired by them, do not make these works the assumed standard by which everything is judged or made. Basically, you should be able to adapt races and classes to a particular setting with no problem. If you want something more specific or more flavorful, that is when you shell out money for supplements. I'd replace Chapter 6 with a longer introduction that goes into things like this.
Rule-wise, it's very hard to keep track of a lot of things in D&D because so many factors depend upon so many other factors. Yet, I would like to keep many of the options skills and feats open up, or even open new ones in the form of talent trees for classes instead of rigid class abilities (my ideas for those are somewhere around here). I don't think skills, feats, or more options are the culprit, but how the rules interact with each other. It seems that everything you want to do beyond "I hit it" and "I keep watch" has a different set of rules that unnecessarily complicates the game. Taken individually, these exceptions aren't bad, but when you try it all at once, it can be daunting and tedious.
The first thing I would do is give a standardize bonus and penalty scale. That way, it's easier to make rulings on the fly and to keep things simple (great for new DMs and players). The bonus/penalty scale I have in mind is something like: minor (+/- 1), moderate (+/- 2), hefty (+/- 5), and extreme (+/- 10 or higher). You can stack different bonuses and penalties together, but the total represents how much you are helped or hindered in a particular action. So, you can have two moderate bonuses and a hefty penalty, resulting in a minor penalty to an action. It keeps things flowing more smoothly than stopping the game, looking things up in the book to see if there's a rule on it, applying said rule, and then seeing what happens.
Skills are relatively easy to simplify. Consolidate some skills, eliminate others, and get rid of the class/cross-class rules. Just give classes a certain number of skill points, which they can distribute as they please, but make the minimum 4 + INT modifier and the maximum 10 + INT modifier, x4 at 1st level. I really hate how D&D penalizes characters for having interests and abilities beyond their niche, and this would do a great deal to overcome that. Granted, this leaves things open to min-maxing and stepping on toes, but I see that more as a courtesy issue than a game balance one.
Feats, rather than making things possible (like Cleave, Great Cleave, Spring Attack, and Whirlwind Attack), should make them easier to pull off or give tangible benefits (like Improved Critical, Spell Mastery, and Item Creation feats). They should also be worth what the character paid for them. At the very least, feats should either give a bonus or offset a penalty. Item Creation feats should be based more on how the item is used rather than what it looks like (more like Arcana Unearthed in this sense). I find it odd that crafting magic items has no Craft prerequisites. I have no idea how to fix metamagic feats, but the way they work now doesn't thrill me.
Combat definitely has to be simplified. There are too many action types, most of which reward thinking only on the tactical level. It's OK if you like tactics, but the topic is streamlining D&D. Going back to the bonus/penalty scale above, what the specific bonuses and penalties represent is up to the DM. In other words, if you want combat to be more tactically-oriented, the bonuses and penalties can incorporate things like preparation, planning, and environmental factors. For more cinematic combat, bonuses and penalties can represent things like style, imagination, and guts. You can also apply both standards.
I'd replace the different types of combat actions with an action slot system, whereby each round is composed of a certain number of slots, and particular actions require a certain number of slots. My default idea is 5 or 10 slots per round, but that's neither here nor there. This makes combat seem more continuous since you aren't taking whole chunks of each character's actions all at once (at least, not unless you want to), but breaking them down into smaller parts. The benefits of this are that combat flows more intuitively, it's easier to adjucate bonuses and penalties to non-combat actions (and a few combat actions as well), and you don't need a 30-second rule or the like to make sure players think quickly and pay attention during combat.
Last, but certainly not least, be rid of attacks of opportunity. Please. Pretty please.
Magic needs a lot of tweaking too. Each spell has a specific effect which affect creatures, objects, or the environment in vastly different ways. In effect, each spell is a rule. This makes the Vancian system necessary just to keep track of things. As with bonuses and penalties, there should be guidelines on what a spell should be able to do given the level and school. This makes is much easier to create your own spells, so listing so many in the PHB becomes unnecessary.
As for the spells themselves, I think introducing spell chains or spell themes would be a good idea. Or rather, make specialization the rule rather than the exception. Speaking of specialization, I think replacing an outright ban on opposed schools with a spell failure percentage (say, 10% per level) or an increasing penalty to a casting roll (see below) would keep options open while adding an element of risk. I'd like to have more enhancing spells than outright overt magic, especially at low levels, so it'd be easier to adapt the system to a game with different amounts and types of magic.
Rather than place the success or failure of a spell on how well somebody else rolls, making spellcasting into a casting check (basically a level check modified by the casting attribute and miscellaneous factors). For spells used in combat or against creatures, this basically reverses the Spell DC rule (except that, obviously, casters must roll higher than a creature's SR to even begin to affect it). For non-combat spells or spells used with objects, it changes things considerably. Since all spells are not cast successfully by default, there is less of a need for the spells/day rule.