There's no such thing as starting from zero
Great news! You're already a polyglot! Seriously, if you can read this post, then you can understand and say (albeit with an accent) words in hundreds of languages!
It is impossible to say that you don't speak anything in a language. Any modern language spoken by a considerable amount of people (I'll exclude tribal languages) has words that you know in it. I don't care if it's Japanese or Swedish, Tagalog or Portuguese, you simply cannot start these languages “from zero”. You already speak some.
No language is an island – they are influenced by politics, technology, trends, religion, history and many other things that us humans tend to share internationally. Since a lot of these topics overlap across the world, you simply cannot avoid seeing recognisable words.
What you already know
Here's a familiar example:
In Italian you can turn on your computer, and in (Brazilian) Portuguese you move the mouse, in Russian you connect to the Интернет (exact transliteration of Internet, И=I, н=n and р=r, can you read it now? :)), and in Japanese you check your Eメール(second part transilteration, “me-ru”, their way of approximating the pronunciation of “mail”), the name of the program you use to surf the net in Turkish is Mozilla Firefox, and you may use Microsoft Windows in the Somali language or perhaps Linux in Euskara to do this.
This gives you hundreds of words before you even start – commonly used brand names, music styles (rock, jazz) and notations, some food and several other categories tend to simply be the same! These words are either untranslatable names, or are originally English, or are originally another language that we happen to use in English. The Czech word robot, is used in most languages, Italian food (pizza, pasta, gnocchi) and music (allegro, forte), or words native to a country, like piranhas in Brazil are just a small sample of the many examples!
Granted, you usually have to pronounce them slightly differently, but saying them to someone with an English accent or reading them when printed will usually lead to little or no confusion.
But wait, there's more!
That's all well and good, but I claimed to be able to teach you thousands of words, not hundreds. The above words are extremely limited so that still leaves you with a lot of other vocabulary to learn. Well, if you happen to be learning a Romance language like French, Spanish, Italian, Portuguese etc., then you are in luck! You can instantly learn thousands of words 🙂
But first, we should consider the invasion that led to the Norman Conquest! In 1066, the Norman Conquest of England meant that an invasion from France was to have French speaking royalty and aristocracy in power in England for centuries.
The once dominant language precursor of English was originally a purely Germanic language. This gives us words like Hand and Arm, which are exactly the same in German and Dutch, and also pretty much gives us most of the language. This gives English speakers an extra edge when learning languages in this family and the similarities greater outnumber the differences.
However, with Anglo-Norman (a precursor to French) in place, upper society in England was basically speaking French for about 300 years. The lower class continued speaking in English of course. This gave a very interesting influx of French words to English, in formal contexts.
Think of a formal version of your word
This means that if you can think of a more formal way to say something, it is quite possible that that is the way to say it in French. Since French is closely related to the other Romance languages, you will find that word will be the same (with a very slight change in spelling) in Spanish, Italian, Portuguese etc.
So if someone knocks on your door, you can tell them to come in, or you could say enter. French: entrer, Spanish: entrar. If you want share your thoughts with someone to show them your point of view, you could also share your opinion and show them your perspective (Italian: opinione, Portuguese: perspectiva) (although here, “point de vue” in French is also not far off!). Instead of showing someone a city, you could be their guide (same in French). Learning vocabulary could be easy, but it's better when it's “simple” 😉
The English language has more words than most other languages and this is because of this influx of vocabulary and large amount of synonyms for basically saying the same thing, but in formal vs informal registers. One huge category of words that are common across English and the Romance languages are the -tion words. Action, nation, precipitation, solution, frustration, tradition, communication, extinction and thousands more -tion words are exactly the same in French; albeit with a French pronunciation. It's easy to change them to other languages; Spanish is -ción, Italian is -zione and Portuguese is -ção.
There are thousands upon thousands of examples
That's just one word ending. There's also -tude (like gratitude, magnitude), -sion (explosion, expression), -ment (encouragement, segment), -age (garage, camouflage) and loads more. Granted, there is the occasional false friend (preservative is an amusing one), or the meaning may be subtly different, but in general you really can rely on this to vastly increase your vocabulary in a nanosecond! Presuming you also know the basic pronunciation system of that language you can also say it correctly!
These terms are known as cognates and one of the first things I do when I learn any language is find out what these are in the language. About.com has excellent extensive sampling of 1,700 such words in French, most of which are the same in other Romance languages. After some practise you will get used to the word endings and know when an English word is almost certainly the same in your target language!
When you see that list, you can tell that it is huge (but still not exhaustive), however that's only the true cognates (i.e. not even a single letter is spelt differently). If you are flexible enough to see what the word looks similar to (exemple, hélicoptère… porto, capitano… astronomía, Saturno, etc.) you can bring that number to tens of thousands of words!!
With this in mind, I like to correct someone when they tell me that they know “no” French, or Spanish etc. and even I have a nice headstart of a few hundred words on Asian languages I haven't even gotten to yet! You have already done the work in learning this vocabulary. If a journey of a thousand miles begins with a single step, you have already taken several steps in the right direction for learning your language 😉