When I'm mining Obsidian I keep either missing the broken block and watching it burn up in the lava, or dying when I try to catch the broken block. Why? I keep falling in lava below – is that the problem? I'm new to Minecraft.
Don't mine the block under your feet. Lava is bad for your health. It also burns up items, including the obsidian you're breaking.
The Minecraft wiki entry on Obsidian has a helpful guide on how to harvest it without either you or the obsidian blocks falling into the lava. In particular, this paragraph:
Natural obsidian "lakes" provide a great deal of obsidian, but mining them can be somewhat hazardous, because there is likely to be lava remaining underneath the obsidian surface. However, the danger can be minimized with a bit of forethought. First of all, fence off any water flows so they won't push you around, especially if the water flows to the edge of remaining lava. Then, look for the edge of the obsidian "lake", or at least a convenient point to begin. Dig a one block deep hole on a level area next to the obsidian, or even in it. (If there is lava under the hole, you may lose the block you just mined, but this is minor.) Place the water source in this hole — if there's lava beneath, place it against the side of the hole, and it will solidify the lava.) Now mine your obsidian, moving outward from the hole. When you expose lava, the water will quench it so quickly that the lava usually won't even burn your newly-mined obsidian. Continue mining outward from the spring as far as the water will travel, after which you can move the water. This method lets you stand on solid blocks rather than in running water.