Why?

Even though green coffee beans tend to be heavier due to the higher water content, generally it's cheaper to roast your own compared to buying them pre-roasted.

You can roast the same beans at different levels to get some variety without having to go out and buy a new batch.

It's kind of fun and a decent conversation topic.

Notes

Don't be scared by how long this post is. It basically just comes down to spread beans on a cookie sheet, put in preheated oven, wait around 12-15 minutes and then take them out and cool them.

Since we're talking about roasting beans, naturally you're going to need a grinder to actually use them.

The process will create some smoke, even with a light roast. Basically, darker roast, more smoke. So far I've mainly done pretty light roasts and even though my kitchen doesn't have much ventilation (and my oven doesn't have fancy modern contraptions like, you know, a light or a fan) it hasn't been an issue.

Your oven should be reasonably clean if you don't want the roasted coffee to taste like random stuff.

If you're a super coffee snob and it has to be perfect, this may not be for you. It's pretty easy, but odds are the first few tries aren't going to be perfect especially if you like darker roasts.

You're going to want something like a large metal mixing bowl and colander for the cooling process. My colander is plastic, so you can probably get away with that if you don't put the red hot beans in it directly out of the oven.

You'll also probably need access to an outside area where bits of coffee chaff blowing around aren't going to bother people. I don't think there's really an easy way to deal with coffee chaff indoors.

By the way, don't try to grind green coffee beans in a normal grinder. They are insanely, and I mean insanely hard and tough. You'll destroy your grinder unless it is an absolute tank. (I'd say it's also not really worth trying, green coffee didn't taste very good to me.)

How

Here's the process:

1. Start preheating your oven to 500f/260c. (Some people say as hot as possible, some people use a slightly lower temperature like 460-475f.)
2. Get a cookie sheet ready. Just a standard cookie sheet. Mine aren't super clean so I put a layer of silver foil on it. Don't preheat the cookie sheet itself.
3. Measure out about 1 cup of green coffee beans. (I've found you can fit about 2 cups on a single sheet but it's probably better to start small.) You want to make sure the beans are spread out evenly in a single layer.
4. Look for beans that are discolored/damaged and toss them away. Don't be a perfectionist though, just get rid of 10-15 of the worst looking beans. Something like that.
5. Place the cookie sheet in the oven once it's reached the correct temperature. I put mine on the bottom rack near the (electric) heating element. If you're going for a darker roast, I guess this might make burning them more likely.
6. Set a timer for ~12 minutes. I wouldn't recommend roasting longer than 14 minutes your first time.
7. Now you wait a bit. Probably around the 8 minute mark, you're going to start hearing sharp cracking/popping sounds. Don't worry, the beans won't jump around like popcorn and the sound is fairly loud so you're not likely to miss it. At this point (or in 1-2 minutes) you can remove the beans and have a light roast. This point is known as the "first crack".
8. After a couple of minutes, the sounds will die off and you won't hear anything for a little bit. If you keep roasting, you'll start to hear a softer, more muted crackling sound start. This is the "second crack". I would not recommend roasting past this point until you're comfortable with the process and have an idea of how roasted the beans are at this point. If you roast much longer, it's very easy to burn them and there's also going to be a lot more smoke.
9. Remove the beans from the oven. You can let them rest for 1-2 minutes on the cookie sheet if you want, then transfer to something like a metal mixing bowl. It has to be something that can deal with 500f stuff touching its surface.
10. Ideally get another mixing bowl/colander/whatever as well. Pouring the beans back and forth through the air is a good way to cool them off and remove chaff. What's chaff you ask? The beans are coated with a papery layer of chaff. Don't worry though, once they're roasted it's really easy to remove. You want to try to cool off the beans pretty quickly at this point.
11. Go outside and blow gently on the roasted beans in your bowl. You should see a bunch of super light, papery chaff fly out. You can pour the hot beans from one bowl to another, and if there's a bit of a breeze that'll help a lot. Otherwise, you can just blow on them. You could also stir them around with a wooden spoon or something to encourage the chaff to separate.
12. Once the chaff is mostly gone (it's fine if there's a little left, or little pieces stuck to some beans) and the beans are fairly cool you can just leave them in a safe place for around 12 hours to fully cool and vent CO2. Don't put them in a sealed container for the first 12-ish hours.

Conclusion

One thing to note is you don't want to actually grind/use the beans for at least 12 hours. It might seem unintuitive, but from what I've read as freshly roasted as possible isn't necessarily best. Depending on the beans/roast level, the coffee might reach its optimal tastiness even a couple weeks after roasting.

I'm far from an expert, but feel free to ask questions in the comments if you want. I can recommend a grinder/beans to get started with if anyone needs information like that.