First things first: there are many, many words out there that have been borrowed into a wide variety of languages, making them almost universal in some parts of the world. Orange (the color), for instance, is recognizable in many of the languages of Europe, and the name of the fruit is common to a smaller but still significant number. Pineapple is almost always a variation on ananas in the West (with English being a major exception). Tea is some variation on tea or cha(i) almost everywhere, and many (although not all) scientific and medical terms are also in this category. If you get a sense of which words these are, and learn the ones not used in languages you already know, you can start a new language with a surprising amount of “free” vocabulary.
In a similar vein, figures of speech and other expressions are often shared across multiple languages. For instance, French and Spanish speakers (among many others) express their age by saying how many years they have, and language can be grouped based on whether they have a distinct verb to have or use a locative or possessive form with the verb to be instead. You will also run into calques and/or compound words that are formed similary in multiple languages — for instance, Finnish, German and Swedish all call a dictionary a “word book”.
If etymology is your thing, note that words that appear similar and have the same meaning aren’t necessarily related. For instance, the English verb to have and the Latin verb habēre both have the same meaning, and look very similar, but are ultimately unrelated etymologically. However, don’t let that deter you — if the similarity in shape makes one of those words easier to remember, take advantage of it! Learning the etymologies of such words, and the coincidental nature of their similarity, can even help strengthen the words in your memory, although if you aren’t interested in etymology and the technical side of linguistics, this may not be the approach for you.
Now, before you go off and start learning, a word of warning: don’t fall prey to false friends! There are plenty of words out there that sound similar, but have totally different meanings — or, worse, subtly different meanings that will distort your message in certain contexts. When in doubt, always check a dictionary or other reputable source to be sure of what a word actually means. Don’t say you’re pregnant when you’re just embarassed!