TNT can be lit by fire.
TNT ignited by fire
Lightning striking on the ground at Medium difficulties and up causes short-lived fires which may be able to trigger an explosions. I haven't been able to test whether lightning directly striking a TNT on Easy or Peaceful will still trigger the explosion.
Lava also causes TNT to ignite as the TNT catches fire because of the Lava. So does lightning if it strikes your TNT, it also causes the TNT to catch fire and it will eventually ignite.
TNT can also be ignited by a Redstone Pulse send out by a Redstone Torch, Button, Lever, Pressure Plate or Detector Rail. This is also the most common and safest method as it doesn't accidentally burn any flammable materials near you.
TNT can also be lit by left clicking it while holding flint-and-steel. In creative mode you must use right-click, otherwise the block is destroyed.
Another source of TNT triggering is explosions. TNT can trigger more TNT, as well as creepers, ghasts or exploding beds in the nether.
The short answer is, it depends. The long answer is, try reading through the wiki page. The medium answer, then, is perhaps the most interesting…
The radius of an explosion is dependent on three main factors: the "power" of the explosion, the explosion "resistance" of the materials being destroyed, and a random factor.
Explosion power is 4 (approximately four blocks) for TNT, 3 for uncharged creepers, 6 for charged creepers, just 1 for ghast fireballs, and 5 for nether beds. Block resistance varies wildly, from 0 for air and weak blocks like redstone wiring, to 30 for most types of stone, to 18,000,000 for bedrock!
When an explosion occurs, it sends out a large number of invisible "rays", each with 70%-130% of the explosion's power. When a ray strikes a block, it's power is applied to the block's resistance. If the power is higher, the block is destroyed and the weakened ray travels on. If the resistance is higher, the ray is stopped, but the block is weakened against other rays.
The upshot of this is that explosions travel to a maximum of 1.3 times their power in blocks, through air. So airburst TNT will reach about 5.2 blocks from its center. Through solid stone, however, TNT will only carve out a single block on all sides (a radius more than one, but less than two).
Best Answer
It's not possible to break bedrock with any amount of TNT, since bedrock has such a high blast resistance that TNT simply can barely make a dent in it, let alone overcome it completely.
Bedrock has a blast resistance of 18,000,000. (That's 3,000 times more durable than obsidian.)
The explosion math is not straightforward, but a great simplification is that a block is destroyed if its blast resistance is overcome by the explosion power applied to it. An explosion's damage is done by radiating a number of lines out from the centre of the blast, and applying damage alone those lines to blocks at intervals (every 0.3 metres) along the lines. If all of a line's damage is absorbed by one or more blocks it has passed through, no farther intervals are checked.
So, the amount of damage a block of TNT can do simultaneously to a particular piece of bedrock is the sum of the damage applied by all the rays that intersect it. This calculation has already been worked on that page as "The minimum block resistance required to absorb maximum blast force of an explosion happening in nearby air". For TNT under optimal conditions (the TNT is sitting on the block), this is 77.67. That means that a block of TNT will destroy a block with 77 points of blast resistance, but not a block with 78.
Explosion damage appears to not be additive (though the deobfuscated code is unclear enough that I'm not 100% certain either way), in that explosion damage seems to be checked for each individual explosion in a separate run of the explosion code and there's no code to save partial blast damage for later explosions to add to, so simultaneous explosions don't add together. We can see this in evidence by causing many, many explosions on a massive scale, and observing that the bedrock appears to remain entirely intact: