What's the safest way to farm obsidian without burning to death?
Solution 1:
You should try setting up an obsidian farm! It requires at least one bucket, but will go faster the more buckets you have. Consider the following picture:
Surround a 1x7 trough with some safe blocks. It doesn't matter what it's made of, as long as the blocks won't burn away (so avoid wood, cloth, etc). This is the object outlined in green below.
Now you should have a 1x7 depression -- you want to put lava blocks into this. Be careful when emptying your lavabuckets, because if you try to place a lava source block on top of another lava source block you'll empty your bucket, losing the lava. I didn't have lava accessible, so for the picture I used cloth. It is outlined in Red.
After you've got your seven lava blocks, simply place a water block on the rim of trough, indicated by the light blue outlined wood block in the picture. The water will flow over each of the seven lava blocks, turning them into obsidian.
This is known colloquially as "obsidian farming".
Solution 2:
There are a bunch of strategies, but since any way to get obsidian requires lava, first and foremost you're going to have to be careful.
Always carry around a bucket of water with you. Remember that you can place the water on the ground, wait for it to spread, and then collect the source block back up. Use this to turn lava into obsidian before getting to near it. You are much better off turning lava into obsidian and mining it in place than trying to move the lava somewhere else and changing it into obsidian there.
Remember that lava produces light, so while it may protect you from spawning mobs, when it changes to obsidian it won't. Light the area with torches before changing the lava to obsidian.
If you're feeling really brave, you can always make a portal to the nether and collect lava there to bring back with you, if you run out of lava in the real world.
Sometimes the lava is really hard to get to safely. If you don't feel comfortable trying to maneuver your way around, it is perfectly fine to abandon that lava spring in favor of a more accessible one. It is always easier to find lava in a cave system than try to mine it out of a pocket surrounded by stone. Just make sure you mark lava somehow (signs work well) so that you don't accidentally dig there in the future and get yourself killed.
When a lava spring is multiple blocks deep, it can be hard to mine the obsidian because the block may fall into the lava directly below it. Make sure you never mine directly below yourself, and don't go diving after the obsidian hoping to catch it before it falls into the lava. These can be especially tricky to mine. Change additional lava to obsidian as it is revealed to minimize risk.
Most importantly, just remember the main mantra: be careful.
Solution 3:
I generally agree with StrixVaria except for one point. Given how long it takes to mine a block of obsidian and the fact that you need diamonds first, it can be just as quick and easy to move the lava in a bucket where you want the obsidian to be and cast the obsidian there. This way there is practically no threat of falling into the lava or "losing" obsidian to deep lava pools.
Of course if there is no lava near your project, mining the obsidian at the source might be a better option.
Solution 4:
Guys you're all wrong. Find a big pool of lava and turn it into obsidian. Mine one and let it get destroyed, then dump water in the hole where it was. Face in one direction and hold down your mouse button. As you mine a block, it is nearly destroyed but the water flows in first, saving the block and making the lava beneath obsidian so you can walk on it. Go 7 blocks then go back to the start and go in a different direction, always in the water flow. It is definitely the easiest way