Verbs make things happen.
Verbs explain what is happening around us. They also express states of being.
The kitesurfer depicted here elegantly brings to mind a cacophony of verbs that illustrate a smooth set of actions and express a complete trick.
The chassis of a sentence.
The verb, in unison with the subject, is at the heart of almost every phrase or sentence. To paraphrase well-known modern author on language, Stever Pinker: "Verbs are the chassis onto which other parts of the sentence are bolted."
Rarely is a sentence uttered in which a verb is not both present and integral to its comprehension.
In fact, in many languages, the verb can be the entire sentence, as is the case in Portuguese when answering in the affirmative by repeating the main verb in the posed question. (Q: Você quer comer? A: Quero.)
As a result, verbs are to be selected wisely, gathered fluidly and collected abundantly.
For the beginner learner, selecting wisely is of utmost importance, and a simple search for the language you are learning plus something along the lines of "most common verbs" should do the trick for the major languages.
If you are learning a more obscure language, you can always find the most common verbs in your native tongue and then learn the equivalents in the target language. You can also try using language software to boost your learning abilities.
Take note on-the-fly of verbs which you come across frequently and are not familiar with.
Collecting verbs abundantly will quickly put you in better standing in your target language as a beginner, on par with your personal phrasebook.
Some distinctions between languages.
Regarding verbs and learning them, it is important to note the distinctive ways they are used in different languages. By this, I am referring to how the verb can change depending on who it relates to, a simple example being conjugations. However, some languages like Mandarin Chinese, do not conjugate verbs and have only one form. Thus, you'll need to figure out the particulars of your target language and study accordingly.
That said, for applicable languages, learning the infinitive form is often a good place to start. This is the equivalent of "to be" or "to go" or "to have", with the "to" in front of the verb. Spanish glosses for these verbs, respectively, are "ser/estar", "ir" and "tener". The Spanish gloss for "to be" brings up another point: often languages will have more than one way to translate a verb, such as ser/estar in Spanish.
Thus it's not always - in fact is almost never - so clear and easy to simply say "I will learn the basic verbs" because of these linguistic idiosyncracies, but deciding to learn the basic verbs and delving into them is, in my opinion, one of the quickest ways to start understanding at least something in nearly any sentence, written or spoken. For example, you might not understand anything else in a sentence but a form of the verb "to want", but at least you will understand that the sentence is referring to someone or something "wanting" something, which is a far cry more than understanding absolutely nothing.
Twenty-five most common verbs in English.
Here's a list of twenty-five of the most common verbs in English, a good starting point for the action-oriented learner :)
be, have, do, say, get, make, go, know, take, see, come, think, look, want, give, use, find, tell, ask, work, seem, feel, try, leave and call (Source)
Understanding something is better than comprehending nothing.
Being familiar with these verbs in their infinitive forms (if applicable) and, even better, knowing them in as many tenses and modes as humanly possible, will open up a world of understanding and allow you to comprehend much more of what is happening around you. Even an incredibly basic knowledge will allow you to ask gramatically-incorrect-but-quite-effective questions, such as pointing to a loaf of bread and then to the person in front of you and saying "to want".
This might not be very elegant, and certainly should not be relied on in the long run!, but it is nonetheless a way of making yourself understood when it counts. It is also a way of understanding something rather than nothing of what is going on around you.
Otherwise related or consumable.
A Steven Pinker video on language and thought.
Use this list of the thousand most common verbs in Portuguese as a reference for learning any language as a list of common verbs to focus on.
Some language hacker suggestions from Tim Ferriss on how to learn a language in three months.
This article's photo of a kitesurfer doing a trick demonstrates a few verbs quite clearly.