After taking the butter out of the fridge, it's cold and is very hard. How can I spread the butter while it's cold and hard, or melt it in order to spread it?
Warm the knife
Ever heard the phrase "like a hot knife through butter"? Warm the knife by dipping it in hot or boiling water, and the butter will be easier to spread because it is warmed on the knife.
Pre-spread the butter
Take some butter and use the knife to spread it about a bit on the inside of the lid of the butter container. (If it's really hard, start by squashing it with the knife.) Then scoop it up and spread again. After a few repeats the butter should be soft enough to spread on your bread.
You can see a (short - 37 seconds) video demonstrating this technique here.
This is the technique I personally use and it works great. I actually use the block of butter itself to spread on, rather than the inside of the lid, but either works.
Use a cheese grater
Graters are designed for grating harder objects, like cheese. They work perfectly on hardened butter, and the grated strands of butter are thin enough that they can be easily spread afterwards.
There are even fancy knives you can buy that grate the butter as you go, if you are so inclined:
Warm the butter
I'm assuming you don't have time to leave the butter to soften at room temperature or you wouldn't be asking the question. However, you can put a small block of butter in the microwave for a few seconds (1-10 depending on size of butter and microwave power), which should be enough to soften it so that it can be easily spread.
Apart from just warming the butter, using cheese plane like this helps me:
It is quite a nice tool to cut the hard butter (like you do with cheese).
Grate your butter onto waxed paper or a cutting board using the box grater like this:
It will quickly become room temperature.
Place the butter between two sheets of waxed paper or parchment paper and shape it with the palm of your hand, or roll it with a rolling pin until it reaches the texture you want.
(Some of advices from this article)