My favorite changes are probably the step into 3D (of course), the heightened emphasis on mini-games, and the more customizable function of Link. 3D gaming in Zelda introduced a world of new possibilities, from beautiful landscapes to more intense combat to improved item control. I do wish they'd slow down and make another 2D game every once in awhile, but the transition to 3D was definitely a turning point for the better, if at least for the concepts it could bring; some may view it as the start of Zelda's downfall, but you can't really say 3D was bad in and of itself without being highly biased.
One thing I have against most JRPGs is how impersonal the world usually feels; you run around, kill enemies, and witness cutscenes. That's usually the bulk of the game, and there's little difference or flair to each to make those categories stand on their own too much. The Legend of Zelda, while not a typical RPG, does well by including numerous mediums of entertainment outside of running, killing, and cutscenes. Want to try your hand at archery? What about horseback archery? Then you've got fishing, skydiving (at least I liked Dodoh's High Dive...), racing as a rolling ball, racing on a horse, racing on foot (or trying to beat some cheating running man, anyway), a Battleship-wannabe, tactically launching cannonballs into barrels, and the famous Money Making Game. These are just a handful of the available mini-games throughout the Zelda series, and the higher emphasis on them is something that should both be continued in Zelda and imitated by other franchises.
Link is supposed to be a "link" to the player, so enhanced customization is a bonus. Link's Awakening DX gave you the optional choice of a Red or Blue Tunic that would increase your offense or defense respectively. The Minish Cap supposedly features rings to provide Link with different bonuses (I'll play it one of these days, give it time....). Skyward Sword was an even larger step in the right direction with nearly everything upgradeable and numerous Medals to affect gameplay. This should be continued, especially with heightened resource management and Medals/Rings that subtly change the way the game is played.
Some of my least favorite changes are increased linearity in dungeon order, enemies being treated like puzzles or simple cannon fodder (or both), and a general lack of post-game content/upped difficulty levels. Zelda is meant to be a big adventure through the wilderness, finding the way on your own - as Captain Barbossa says in Pirates of the Caribbean: At World's End, "For certain, you have to be lost to find the places that can't be found." I'm not necessarily saying every Zelda game should be like this, though the series probably wouldn't suffer much if such was the case, but it's something that should be brought back frequently. I have an entire article written up on how such a feature could work in contemporary Zelda, and it's more than possible. Zelda has now been reduced to combat and puzzles, which were originally supplemental to exploration, not independent from it.
Most Zelda players acknowledge that enemies have been lacking recently, and it's because they're usually being treated as either puzzles or cannon fodder. Puzzle enemies become redundant (they'll be ruined or destroyed!) when you know the answer to the issue they present, meaning they quickly lose their luster. Cannon fodder enemies were prevalent in Twilight Princess. You know what I'm talking about; the hordes of Bokoblins or Lizalfos that just hang around you doing almost nothing, waiting to be killed en masse. Combat in old 2D Zelda games was incredibly simple, yet also very deep. Torrents of enemy projectiles could be dodged with clever sidestepping, and some foes required hitting at an angle to damage. They were aggressive and cunning. Now it's usually "Hit the enemy where it's not currently guarding before it retaliates ten seconds from now!" Not terribly challenging. Novel, I'll give it that, but lackluster after awhile.
Older Zelda games also featured post-game content, usually in the form of the same game, only harder and swapped around. This was a staple of a lot of older games, and it's something simple that modern Zelda games would benefit from. Hero Mode in Skyward Sword was a great step in the right direction, but it's one of the first true post-game challenges in Zelda for years. I'm talking about increased enemy attack, health (possibly), presence, intelligence, color/costume swaps, and other fun gimmicks. Most Zelda games these days end for good at the final boss, which isn't terribly useful for an action RPG.