Is it better to vent a bathroom fan out the soffit, the side of the house, or the roof?
As far as I know, it doesn't really matter as long as it's vented outside. Venting into the attic is Very Bad -- in the winter, the humid air will condense and (if cold enough) freeze, and you'll effectively have water in the attic. 

For any vents, the straighter the run is, the more efficient and quieter it is. Avoid corrugated pipes, and avoid bends and elbows as much as possible. For that reason, going into the soffit almost guarantees you'll have more bends than going through the roof.


As an aside, I have two bathrooms, and they both have the identical fan (which is rated as very quiet). One bathroom has a 3" duct that is about 3' long, and goes almost straight out the side of the house (it's done with a small section of flex line that bends up and over be a couple inches, because I lined up the vent on the outside with the siding). The other bathroom vent has a 4" x ~6' line that has two sweeping turns and goes out the side of the house. The second one is noticeably noisier than the first.


As a DIY job, you'll also have to be sure you have the knowledge to cut a hole in the roof. I'd say it'd be far better to go out the side of the house (which is a bit easier to seal) than to go through the roof and cause a leak.