I have to say that there is a LOT of work needed on teaching when classifiers are necessary and when they are not. For example, why am I not needing to use cái when I talk about thunder or clouds? Why don't I use it when speaking of xe đạp? The notes indicate that cái is used for 'almost every object', but that's obviously not the case. There is a lot of confusion in my mind as to when classifiers are necessary and when they are incorrect.
At this point in my learning, the use of con and người has not been problematic, but my experience thus far is just leaving me to wait for the other shoe to drop.
In short, I think there is a lot of 'use cái when you learn to use cái and not when you don't', but that doesn't help to understand the underlying structure of the language and could definitely lead to problems down the line (depending on how the course deals with the matter).
Just to clarify, the note says that in most cases nouns "can" be accompanied by cái. I take this to mean that if you have a noun, then there is a good chance it will be associated with cái; I do not assume this to mean you will use cái with that noun all the time. Maybe the wording there could change to make it more clear?
I'm certainly no expert on when to apply classifiers but I will leave a few resources here I have been using:
Linguistics of Vietnamese: An International Survey is a book chapter that delves into the subject, though not all the pages are available on Google.
Loại Từ – Classifiers (PDF) is an article that briefly describes classifiers and then provides a great list of them with examples.
A corpus-based analysis of Vietnamese 'classifiers' con and cái (PDF) goes into even further detail, specifically with con and cái. This is an academic paper exploring use in practice.
A Vietnamese Reference Grammar is a book that provides a perhaps more succinct explanation of the use of classifiers: they most frequently appear in numerative phrases (e.g. một con mèo), and serves the role of specifying an individual unit. This is important since nouns in Vietnamese do not inherently include this property.
To add to the confusion cái can apparently serve a role similar to but different than other classifiers, though that is a more complex grammatical function you could probably pass on at the start.
I am still learning when, how and when not to use classifiers in general, but there is some helpful rules in the book Get Started in Vietnamese by Dana Healy. I'll write up the section and post it here. Note: cái is generally used to categorize a "thing". That literally what is means - "thing". It's mainly used for inanimate objects, so you can see why cái can be ubiquitous in its use. :)
Here's an excerpt from the book. Hope it helps. :)
"Classifiers are grammatical words used in Vietnamese to specify nouns and 'classify' into which category they belong. Nouns in Vietnamese can be used either with or without classifiers according to the following rules:
1: If you are making a statement which refers to a specific object, person, animal, or activity, then you must use a classifier. If the reference is general, the classifier is omitted. For example: Con chó của tôi nhỏ - My dog is small (The reference here is specific, you are referring to a specific dog, i.e. your dog and therefore a classifier must be used.) Tôi thích chó - I like dogs. (There reference here is general. You are not referring to any dog in particular, but to dogs as a species and therefore there is no classifier.)
2: Classifiers are always placed immediately before the noun they classify. No other grammatical word can be inserted in between a classifier and the noun which is it linked. Cái nhà (a house), con mèo (a cat), cuộc họp (a meeting) etc.
3: When a statement refers to a certain number of objects and people then classifier is normally used: ba con chó (three dogs) hai quyển sách (2 books).
4: Classifiers can be omitted when it is already clear from the context that the reference is specific."