I took a while to realize the meaning of the line "Rangers at the core" in the Power Rangers Zeo theme song - I thought it might have something to do with the Rangers in the cockpit of the Megazord, or perhaps them being inside their suits, but eventually I realized that what it actually meant was that even though the Rangers had different powers and equipment, they were, at the core, still the same Rangers continuing their fight against evil in the tradition they'd followed in the Mighty Morphin' series.
Tommy's sudden boost in intelligence and memory skills (as seen In Dino Thunder) may be due to Prince Gasket messing around with Tommy's brain in "King for a Day."
Rita being pregnant with their child by the events of the season finale as per Word of God explains quite nicely why she and Zedd left at the end of the season, rather than resume conquering — they were taking time off to raise their son.
In addition they may have gone off to another planet that wasn't as well defended to build their forces up so they'd have a better footing when they came back.
If one really thinks about it, the Zeo Rangers as described in the series canon are, literally, the most powerful rangers of all time. The Zeo Crystal was an infinitely self-strengthening power source that would never stop growing more potent. The Zeo Ranger arsenal of weapons and zords was growing more complex and varied with every major episode. But this leads to a serious question that was never answered in any way. Why did they stop using the Zeo powers and move to the Turbo powers? Tommy at the least clearly retains the Red Zeonizer all the way up to Forever Red and it's unlikely he got rid of it afterword, unless that was the base he used to develop the Dino Gems. But that also begs the question of "Why mess with the Zeo Crystal and turn it into something clearly less powerful?" The only answer that logically comes to mind is that the Zeo Crystal is not as much of a penalty free power source as first believed. As we see with season team-ups and was Lampshaded in Power Rangers RPM, morphing tends to cause explosions. Imagine what Zeo morphing would eventually do. That's right, the Zeo powers were pocket supernovas waiting to happen.
Keep in mind that these crystals were split from one single crystal... Imagine the power that would cause if they were still united.
It's also possible that when Katherine's morph failed in the Turbo movie, the whole Zeo powerset was damaged or incomplete, and it just couldn't be repaired before the old team left.
Another possible explanation is the Zeo Cyrstals are the power source for the Turbo Powers. This would answer the question of why the Turbo Powers are supposed to be more powerful then the Zeo powers. They're not, they're just more efficient, using less power for the same effect.
One thing to consider: It's stated that the morphing grid gets its overall power from the balance between good and evil. Perhaps they had to stop using the Zeo powers or risk destabilizing the whole morphing grid due to a shift in the balance. This also explains why each season of rangers stops being rangers when their evil bad guys get defeated, and when the monsters get bigger and badder, so do the rangers. A subtle, justified example of Mid-season upgrades, available for every season ever.
It didn't make much sense why, after the Command Centre blew up, the rangers found the Zeo Crystal abandoned, and why in the first half of Zeo, Goldar and Rito had lost their memories. But with some further analysis, it makes sense that it was because of the forcefield around the Zeo Crystal.
It had been established in Season 3 of MMPR that a forcefield had been placed around the Zeo Crystal by the inhabitants of M51, to destroy anything evil that comes near it. It had been believed that it would destroy anyone who was evil, or who was ONCE evil, but when Tommy, who had been the evil Green Ranger, took the crystal, it was clear that the forcefield did have an effect on him, but merely removed any remaining evil that was in his system.
It was also made clear that this forcefield remained around the Zeo Crystal even after it was removed by Tommy from underneath Zedd and Rita's palace, because Tommy used it to cleanse all the evil from Kat.
So when Goldar and Rito broke into the Command Centre and stole the Zeo Crystal, the forcefield acted on them, removing all the evil from them, including their memories of ever having done any evil. This explains why Goldar lost his wings (they were given as a present by Zedd, presumably made of pure evil), and also explains why Goldar and Rito never had any desire to do anything evil while they were in this state, as all the evil had been cleansed from them.
Unfortunately, Zedd and Rita caught up with them, and in an even more prominent form of Fridge Brilliance, it makes sense why Zordon's Wave turned Goldar to sand when he wasn't destroyed by the Zeo Crystal's forcefield. It's because when Zedd and Rita found Goldar and Rito, Zedd agreed to restore Goldar's memory, wings and sword, on the condition that Goldar never do anything good again. So, in making this agreement, Goldar literally stripped himself of any remaining good that was in his system, which is why when Zordon's Wave hit, there was nothing to save.