Saving enough money for anything isn't hard. It just takes discipline and will. For these discussions I have to assume you have a somewhat steady stream of income.
Don't Borrow if you Already Owe
If you already owe money on credit cards, TVs, appliances, or other "stuff," then pay all of that off first. Whereas I view taking trips like these as investments - investments in education and understanding of the world's problems -- you don't need to be making investments if you are paying off debt. Get yourself out of hock first.
Now don't take this advice as a trip-killer. You can use a once-in-a-lifetime vacation as a reward for getting out of debt. And as you see yourself making progress every month you can simultaneously be planning that big adventure!
Cut out the Fat
Saving money is the opposite of spending it. You need to stop spending money if you want to save. And that means eliminating all of the superfluous junk that your money is funding today. Make a list of everything you spend money on and put it in 2 columns - what you need to survive, and everything else.
In the first column you'll find things like groceries, rent, and electricity.
In the second column you should find things like cable TV, coffee, alcohol, cigarettes, movies, new clothes, desserts, magazines, newspapers, restaurants...
Are you surprised at what I listed in column 2? Well you shouldn't be. You don't NEED any of that stuff. Calculate or estimate what you spend each month on all of that junk. Take a big red pen and draw an X through column 2. You have just resolved to stop spending money on crap you don't need!
Now look at each item in column 1
Put a Little Away Each Paycheck
You know that calculation you made about how much you are spending on junk each month? Go open a brand new bank account and arrange to have that much of your paycheck automatically deposited into it.
IMPORTANT -- IMPORTANT -- IMPORTANT -- IMPORTANT -- IMPORTANT -- IMPORTANT
This bank must be out of your reach. You must not be tempted to siphon from this account. Choose one on the other side of town. DO NOT get a credit or debit card. Just checks.
IMPORTANT -- IMPORTANT -- IMPORTANT -- IMPORTANT -- IMPORTANT -- IMPORTANT
If you owe money on credit cards or other debt, use the money from your new account to pay the minimum balance on ALL BUT ONE of your cards. That one card should be the card with the lowest outstanding balance. Then take all but $20 and put it towards paying off that lowest balance. When you pay off that card, cut it up and throw it away. Repeat until the last credit card. Don't destroy the last card after you pay it off. You'll need it on your trip.
When your bills are paid, keep the account and keep the money flowing into it. This is now your vacation account. Although I recommend that you fill it entirely with all the funds you believe you will need for your trip, it is not necessary. If/When you get this far you will have proved to yourself that you have the discipline to pay things off, so you should be able to start booking your trip before this account is full. Definitely wait until your bills are paid, though.
Your Advice Sucks, I'm Gonna Be Miserable!!
The happiest people on earth are those who are the most grateful and the most productive. Start by being genuinely thankful for what you have. I literally believe that Thanksgiving is every day for me.
Then start being more productive. Take all that extra time you might find on your hands and apply it to learning a new language. You might need that skill in the foreign country you're going to visit. Spend more time with your friends and family that you've drifted apart from.
Take up a free sport like running, volleyball, or basketball. You will find yourself accidentally getting in shape.
Or get a second job, but make it a fun one. Go work in a camping store or a movie theater at night. Find a job that might teach you something you might need to know on your big trip.
Guess what? If you take a second job temporarily you can apply that entire paycheck to your new account. And that means paying off your bills even quicker.
Save Enough, Then Quit!
One option that only a few people ever consider in their lives is to work really hard, save a lot of money, and then quit their job. Now they have way more than 2 weeks of vacation time - they've got as much as they've saved and planned for. If your profession is in high enough demand, you might consider this option.