What's the Explanation?
For one reason or another, irregular verbs cannot keep their infinitive bases in some or all of their conjugated forms. What are the reasons behind these irregularities? Some irregularities are actually “regular”; that is, the changes that verbs undergo run consistently across verb groups. You will see that group irregularities also depend on similar letter substitutions.
Some irregularities result from spelling accommodations describing changes that occur to keep pronunciation consistent. A few verbs are “naturally” irregular (most often these are words that have been present in Spanish for a long time, and have changed radically over centuries; ser is a good example—it is irregular in most of its conjugated forms). Fortunately, only a small fraction of all Spanish verbs are irregular.
As you learn each verb, check to see whether it is regular or irregular. If it happens to be irregular, try to see if its irregularity is the same as that of another verb you are already familiar with—then, all you'll need to do is memorize how to conjugate one irregular verb instead of two. For example, did you know that estar and tener behave similarly in the preterite tense?
A Good Strategy
Keep a notebook with a list of the irregular verbs you think you will more than likely need. At first, limit them to about ten verbs throughout all tenses, but keep adding another verb or two every week. The progress may seem slow, but all you really need to do is become aware of the potential irregularity of a verb. At that point you can do what native speakers do—consult a Spanish verb manual or an online verb-conjugation search engine. Also available for your reference is a condensed verb chart in Appendix D.