First of all, the high-level wizard encounter you're describing is probably among the hardest kinds of encounters to run, for exactly the reasons that you mentioned - the bad guy has lots of options, some of which require a fairly detailed understanding of the rules.
There are a few techniques that I'm aware of for dealing with such things.
Invest the Time
Even well-practiced GMs can spend hours preparing for a complex encounter. For a published adventure, this usually means reading the encounter description and written tactics several times, as well as visualizing unusual scenarios. For an adventure you've written yourself, read over the monster stat blocks, as well as any uncommon rules that are likely to crop up (environmental hazards, uncommon conditions that the monsters can inflict, etc.).
If you know which PCs will be playing the game, you can get a pretty good idea over time of how they think and respond to particular challenges, which can help quite a bit (as opposed to a convention situation, where you have to be prepared for whoever shows up).
Of course, you don't have to prep this hard for every encounter - focus on the ones that are most important to the plot. If the grand showdown against the powerful wizard goes well, the players won't mind if one of his lackeys didn't seem quite as difficult as he should have been.
Some GMs find it helpful to write their own tactical notes that they can refer to during the encounter. This can take many forms depending on what works for you, but some typical examples are:
- Consolidating rules and calculating things ahead of time. If multiple possibilities exist, 5e is pretty good about providing all the numbers (i.e., X damage if using one hand, Y damage if using two), but there are still situations where this can be helpful. For example, if you know the PCs will be fighting duergar above ground, you can note that they have disadvantage on all of their attacks right next to the attack statistics (instead of it being a few paragraphs above where it normally is).
- Listing contingency plans. Enemies like wizards will often have their go-to spells and then other spells devoted to specific circumstances - Dimension Door for ease of movement, Faerie Fire to deal with invisible enemies, etc.
Turn the Encounter Around
If possible, put yourself in the PCs' shoes. Ask yourself what they'll be able to determine about the encounter and what things will only be clear to you, then ask yourself how you'd deal with the challenges being presented. Which enemies will they focus on? Are there things that the monsters will do that will immediately draw their attention, like inflicting certain conditions?