No I think you need an iron pick or diamond (or gold, but there's no point making gold tools they aren't very durable).
The lower you go the more redstone you'll find, but it's quite common in the layers where it's present so no doubt you will find it if you just keep digging down, long before you hit bedrock.
Bear in mind 'lava' isn't the bottom of the world, lava can appear pretty much anywhere although it is more common lower down. You know you're at the bottom when you hit bedrock - a darker stone that can't be mined.