This is an addition to Chistophe's answer. I have not seen the formal list of such verbs. However, from learning through Pimsleur audio classes I noticed that some verbs and words are usable in a wider variety of situations than others. For example:
"understand" is more useful than "speak".
"can" and "want" can be used in multiple situations, e.g. I can. I want. Can you? Do you want?
Pimsleur is using almost the same order of learning words when teaching different languages. This supports your assumption that some words are "more important" than others.
Frequency dictionaries are probably the best existing source of such information. However, I suspect that frequency of words in learner's speech is different from frequency of words in native speaker's speech. As an example, Pimsleur teaches numbers early enough in the course, because they are necessary to converse about time and money and family members. However, specific numbers: seven and eight may not be high on frequency lists of native speakers.
As a beginner learner and a tourist you will likely need to speak about a flight number, passport number, time of the flight and a gate number. But will not need to discuss yesterday TV show, or a hockey game, or your recent favorite book. Native speakers are likely to spend majority of time discussing other/different things.