I've based this reply on someone going to a more remote part of the world. North America or Europe would be a different situation. Also, I only take photos, so video is out of my area of knowledge.
I just returned from 5 weeks in India. Backing up photos was not something that I thought would be a problem before I left. I had what I thought was a fair amount of storage. I had envisioned burning DVD's every day or so and clearing off my cards. I thought I had it covered. Easier said than done.
Depending on where you're going, you may not have access to either high speed internet or fast computers. This means that even if you decide to burn your backup photos, it may take forever. Also burning DVD's is by no means universal, so if you have a 6-10 megapixel camera and are shooting large files or RAW images you will be burning dozens of CD's.
Burning DVD's is not exactly cheap either. It may cost you up to $4 per disc, remember you're at the mercy of whoever is doing this for you. If you're in a small town and your cards are full you might have no other options. If you are duplicating your DVD's you could spend around $10 copying a 4 mb card.
The option that I used was usb memory flash drives. You can get a 4 GB drive for about $15. So if you have 3 or 4 of them you can transfer your cards to them on any old computer and then burn DVD's when you find a cheap, fast internet cafe. The beauty of these are that you can carry them in your pocket and chances are if you get robbed, they won't be stolen. I would suggest that you buy them before the trip and test them several times to make sure they work properly. Some flash drives can be recovered after being submerged in water, but this is not universal.
You could use memory cards / sticks / whatever but they are not as cheap and you probably shouldn't use knockoff brands. They're cheap for a reason. A 4 GB Sandisk CF card will set you back $60. This is the best way to go if you can afford it. You can just use your cards and not have to transfer the photos until you're ready to copy them to DVD. If you have a digital SLR and shoot a lot of photos you will probably need 5-6 cards. The CF card can be recovered if they get wet, even if submerged for long periods.
Another option is portable hard drives. Not my favourite solution. You get a large amount of storage but they're bulky and if you lose it you are SOL.
The last solution is a computer. Nice because not only can you store your photos but you can work on them as you go. BUT it is a risk taking a valuable item into more remote areas. I ran into someone on my travels who brought a computer and dropped it. Lost everything on it. It will probably be retrievable but at what cost.
I ended up taking three 2 GB CF cards and three 2 GB flash drives. Even with 12 GB of storage I still ran into problems a couple of times. Once I ended up buying a 4th CF card, a cheapo which failed and I lost dozens of photos. Another time I spent 6 hours getting CD's burnt because I had no memory left and I was in the middle of nowhere. I resisted the urge to delete files off the camera, always a bad idea. I would trade the $30 that a couple of flash drives would cost to get those 6 hours back in a heartbeat. Worst of all though was the fact that several times I stopped taking RAW images and had to stick with JPG's 'cause I was running out of memory.
My photo storage motto has now become, "Better to have too much than too little."
Travel well and take lots of photos.
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