I've faced a similar problem since I taught C, C++, and Java.
Here is what I do:
First, I keep packages separately and explain them in the end.
Ideally, in my opinion, students should first learn about ADTs, preferably in C. They have the struct, they have the separate operations on it. Fields are then simply the "slots" in the struct and you can even show the memory layout to demonstrate it. Functions are separate entities that operate on those structs.
You then make the transition to classes, methods, and fields and show that in essence (barring inheritance and some anecdotes) they are in many ways syntactic sugar for ADTs.
If you need, you can then teach object layouts, inheritance, and virtual tables (in my experience it helps students understand inheritance better to see the memory layout).
Finally you get to the topic of how to organize classes together. If you teach C++, you don't really have packages but you can explain namespaces and discuss organization and separate compilation.
If you are in Java, then you just explain that these are collections of classes in the same namespace, that have special access rules and show them. The package system in Java is kind of broken anyway so I usually go through patterns (e.g., separating a UI package from the C).
So in summary: Classes form the basis for objects that are a memory arrangement of several fields and associated methods that operate on them. Packages are collections of classes that have one more access restriction mechanism.