Judebert
If you're running a "race around the world", you're going to find yourself in a foreign land where they don't speak English. You'll need directions, or you'll want your taxi to go faster. How will you get your message across?
Previous Racers have tried pantomime, wordless noises, and drawing. Generally, these aren't very helpful: some people aren't good at charades, different cultures use different noises, and pictures are easily misunderstood.
You'll want to speak the local language.
You can't be expected to learn every language on the globe, but each member of your team should understand at least one other language. English is the lingua franca of business, science, diplomacy, and aviation, but it's not an advantage over other teams.
To cover the majority of the globe, you'd also need French, Mandarin Chinese, Portuguese (Spanish is similar), Italian, German, Afrikaans, Swahili, Arabic, Russian, and Hindi. There's an interesting set of language lists that could help you decide which ones you want to take on. In particular, the "number of speakers" and "number of countries" lists would be most useful.
Since you're just traveling, and not actually planning on settling down to live in one of these foreign countries, you don't need to be conversationally fluent with any of these languages. In fact, I've heard that just 100 words is enough to be functionally fluent in any language... but I've never been told what the words are, specifically.
Here are the words I think you should learn for functional Racing.
Civility
Although these should be obvious, I've seen so many rude Racers that perhaps I'd better include them here. Learn to say "please", "thank you", and "excuse me". Use them often. If you've got the time, learn "good morning" and the like, too.
Numbers
This should be blindingly obvious as well. Not everybody uses the Arabic numerals, so don't count on seeing numbers you know. Merchants won't necessarily have posted prices anyway, and passersby will be telling you the directions, not writing them down. Learn to speak the digits, then the multi-digit numbers; for money, you'd probably like to learn everything up to 1000. Then start on the ordinals (first, second, third, fourth, etc).
Directions
You'll want to know the cardinal directions (North, South, East, West), as well as the relative directions (left, right, forward, back) and the turning commands "turn around" and "straight ahead". Learning the relative positions (in front, behind, next to, before, after) wouldn't be a bad idea, either.
Distance
Luckily, most of the world uses metric, so the word you need to know is "kilometers". In a city, you'll need to know "blocks" and "streets". In the country, distances are often measured in time, so you'll want to know "minutes" and "hours". You'll also want the relative distances (nearest, closest, farthest).
Transportation
You'll need to ask for the bus, train, plane, subway, taxi, boat, and bicycle. You need to know the word for "ticket".
Particularly for planes, you need to distinguish between arrival and departure. "Soonest" and "fastest" are what you want, but you'll need to know "latest" and "slowest".
Time
Since you've already learned the numbers, this should be a cinch. We've already covered durations and relative times, too. All that's left are time-related concepts, like "hurry", "slow down", and "stop". "Today" and "tomorrow" might come in handy, too.
Queries
Most important: Where's the bathroom? 
Learn "Where is", along with "hotel", "restaurant", "travel agent", and "airport". You'll get the names for any specific places you need to go, so "where is" should take care of those directions.
You'll also want to know "How much?" You'll be asking this of taxi drivers, ticket agents, and merchants. "How far" is important for directions, and "How soon" for travel planning.
"Where am I?" and "Can you show me on my map?" will be required if you get lost.
"I want" can be helpful. Remember to use "please" and "thank you"!
