Set a budget. Naming your own price is a matter of bartering and sticking to a plan, so make a detailed budget for your trip. This will vary depending on the destination. It's best to break down your travel plans into sections: flight, hotels, meals and then any extras you may be interested in, like car rentals and sightseeing tours. The limits here are entirely up to you, but expect decent hotels in most countries to range around $60 per night, except in Asia where they are considerably less expensive.
Email airlines. Visit the airline websites and find the email address of sales representatives and outline a specific price. Word your emails like this: "I'm looking for a flight to Puerto Rico on January 6, and I'm only looking to spend $250." Then email others. You may not get a response from every airline; you may get a counter offer. The more feelers you send out, the more likely you are to get a response. Being specific helps; airlines can check specific flights and see if any seats remain unsold.
Contact hotels close to your departure date. It's very unlikely that hotels will offer discounted rooms months in advance as full-paying customers may yet book empty rooms, but as you get closer to the dates you plan to stay they'll be more anxious to sell off empty rooms. There are numerous websites that facilitate these last-minute deals, but you needn't rely on them. Contact the hotel directly and offer a price you're happy with. It's a good idea to be aware of the regular rates so your offers are reasonable. A $200 per night hotel isn't going to give you a room for $15, so be smart with your offers.
Don't book ahead. If you're adventurous and carry cash in hand, you can find great deals once you arrive. Be mindful of holidays, which cause rates and bookings to spike, but don't be afraid to try the reservation-less route during off-seasons. Hotel and guesthouse owners are eager to barter during the low seasons, especially in Asia, and showing cash while bartering is incredibly effective.
Barter everywhere, for everything. In North America, this is pretty uncommon but a lot of cultures and countries are comfortable with and use bartering in most transactions. As a general rule, things with a clearly marked price tag are non-negotiable, but not always. In Korea, for example, the price tag is simply a starting point for negotiations. Don't be shy. Most services -- taxis, car rentals, or Internet service -- can be negotiated. Cliche as it sounds, a smile is often your best weapon when bartering.