As a student at Macalester College in St. Paul, Minnesota, with parents living in the New York Metropolitan Area, traveling is anything but climate friendly or inexpensive. Further, as a youth climate movement leader, I usually have a couple conferences a year in far-flung parts of the country - luckily, the sponsoring organizations usually pay for travel, or I couldn't afford it. Even if the price is covered, air planes are so carbon intensive that 2-4 round-trip flights a year is the majority of my carbon footprint. I bike everywhere and otherwise keep my lifestyle carbon emissions very low, but flying wipes out all these gains - and as I get more involved in the climate movement, the amount of travel I do only rises.
Some climate conscious air travelers offset their emissions by buying offsets from certified providers. The best one I know of is TerraPass for Flight, which calculates your carbon emissions from flying and sells an offset pass based on the cost to offset an equivalent amount of emissions through investments in clean energy. The cost for 6,000 miles of air travel is about $10, so it's cheap enough for even a cash-strapped college student (that's probably about as much flying as I do in a year). There are other offsetting companies, but many of them rely on investments in land use changes which are less reliable - though there are probably other good systems. I don't actually use offsetting programs, primarily because it feels to me like paying for the problem without really solving the problem. At $10, it's not un-affordable, but I make a much larger impact by building climate solutions on my college campus and local region in ways that are much more empowering. Air travel is one of the hardest sectors to cut emissions in, and its the one that has been most overlooked in climate negotiations. The primary reason for this oversight is that air travel is an international source, and thus can't be managed by a single sovereign nation. Others have suggested it gets overlooked simply because the negotiators themselves fly so much. More than most other sectors, air travel is facing a real lack of solutions.
I don't have any hints on intercontinental travel, but I have found a cheap, energy efficient, and comfortable way to get around the US when I have a bit of time. Car travel over long distances is no better than flying, so the only real solutions are bus or train. I haven't been able to cut out flying entirely, since I often need to be attending events , but using trains and buses has allowed me to avoid flying when I'm changing 'home' locations where I'll be staying for longer periods of time. I've found taking trains and buses to be much more enjoyable and strangely less time consuming. I'm also saving money doing it!
Amtrak is pretty much the only remaining national train company. It can get you nearly anywhere and is a key link in most non-flight cross-country trips. The train is about as fast as driving, the difference being that you can sleep in your seat or read a book as you go. Train rides are great since you get to see lots of awesome countryside, have plenty of time to do whatever you want (I've finished two books each time I've taken a cross-country train ride), and when you're done you don't feel like you've been jerked across the country at 35,000 feet. One downside to Amtrak is that its often delayed (one trip arrived 2 hours late, while the next was early). This is largely due to the fact that for some odd reason, freight trains get priority on the tracks, which is particularly ridiculous when one considers that nearly 60% of freight in the US is coal, the absolute nemesis of efficient transport like trains. The other downside is that unless you plan it right, Amtrak trips can be pretty expensive, since in many ways its been marketed as a luxury transport option - again, a niche market created when the interstate highway system wiped out trains as the best kind of long-distance transport. Coach class is not unreasonable, but if I'd taken Amtrak straight from St. Paul to New York, it would have cost ~$170 each way, which is more than the $270 round-trip usually available when flying.
The solution is to use a regional bus service, such as Megabus. Megabus covers major cities in the Midwest region, and proved a very efficient way to get from the Twin Cities to Chicago - for only $20. Megabus has a special promotional, where if your the first buyer of a ticket for a given bus, you get it for $1, and the prices slowly rise after that - up to about $20. Most of these super-cheap regional bus companies make their money but getting the bus very full and making few or no stops between origin and destination. If you can start from and end in a big city, there's a good chance that a regional bus service can get you there or at least make the train link much cheaper. By taking the Megabus to Chicago, I could get an Amtrak ticket from Chicago to New York for only $80, making each way only $100 - cheaper than flying.
The other main regional buses I've used are the Chinatown buses operating out of New York, Boston, and Washington DC to surrounding areas. It's $20 for a one-way from NY to either Boston or DC, which is much cheaper than regional train lines or flying. I suspect that there are similar regional buses in most parts of the country. If a bit unconventional, I've found them convenient and efficient.
So what about the carbon? Trains are the ultimate form of efficient transportation: they use fuel, but the low steel-on-steel coefficient of friction and the large number of people per train means that the per-person fuel usage is very low. I've heard that freight trains have an efficiency approaching 1,000 miles per gallon per ton of freight - while the majority of the weight of a passenger train is the train itself, there's almost certainly less than a ton of train weight per passenger. In any case, the fuel used, and thus carbon emitted, is just a tiny fraction of flying or driving. Buses, which have the same runner-asphalt friction coefficient as cars, are less efficient than trains, but because they transport so many more people with just a couple times more fuel than a car, they are still several times more efficient than driving. Air travel is about as carbon-intensive per person as driving if you have a couple people in the car, so both forms are dramatically more climate-friendly than air fare. I've never tried to do the calculation, and I'm not really that interested in it. I still fly too much since I can't spend a day and a half traveling all the time, even if I'm getting good work done. But it's a cheaper, more comfortable, more climate-friendly way to travel - so it's a start.