I could probably write a lot on this, but for now, I'll go with the following:
- planet shots from orbit (the early seasons especially). They really could do a lot with this in making each planet unique (moons, size, etc.) They also could match it up properly (i.e. where the Enterprise revisited a planet and the planet actually looking the same [not only within TNG, but also to TOS - for example, Vulcan]).
- reused shots [as several others have already pointed out]: primarily, the simple shots of the Enterprise flying through space during a captain's log, etc. Use different angles and put more and varied space "stuff" in there [like the title sequence of Voyager]. Also, it would be cool to see the different classes of ships, so unless it's actually spelled out somewhere in an episode, make some of the ships different (kind of got boring seeing the Excelsior again and again).
- space battles. 'Yesterday's Enterprise' is kinda slow-looking now, I think. They really could jazz it up (and all the others) big time. Also, some of the torpodeo shots look quite bad now (again, especially in early episodes). Also, again, angles are important - they reused the same shots for the ship firing, so they could sort that.
- those effects shots were actors get zapped or shot with something. Some of that looks quite bad now.
- in terms of a specific episode, 'Aquiel' comes to mind. The 'shape-shifter' looks really dated - I'd love to see an upgrade on that.
I'm tempted to start rewatching a few episodes now so I can suggest some more upgrades!