I use propane and atmosphere. Sometimes I use propane and oxygen. I tend to avoid using oxygen because it's expensive. Once in a while, I will use a small butane torch, mostly for flame finishing waxes.
I learned on acetylene-atmosphere at my first school. Acetylene is cheap, but very dirty and you need a good ventilation system. When released, acetylene rises, so in that respect it is sometimes considered safer than propane. Propane drops when released so it can pool in corners and in low areas of your house / basement. That's generally a bad situation to have.
If you solder directly in front of an open window, you can place a fan in the window to suck the fumes into the fan and out the window. You can also place an additional fan behind you to help blow the fumes out through the window. However, the fan behind you might bother your soldering work.
Do not solder over the stove. The vent hood on the stove seems like a good idea, but the vent hood actually draws the fumes upward, right up past your face, so you are breathing those in. You want the fumes to be drawn away from your face and out of the building (like through a window).
In crowded cities, I understand they don't let you generally solder with propane tanks because of the fire risk. In San Francisco, you have to hire a plumber to hook up a natural gas outlet from your natural gas line to your soldering table. This is generally considered the safest regarding fire risk. At my second school, I learned on natural gas-oxygen.
Regarding breathing fumes, using a respirator is fine while you're soldering but the fumes are still there when you take it off. If you need to use a respirator, then you know you do not have adequate ventilation. I find it's best not to use a respirator so you can detect by smell if there is a problem (doesn't work for everything) and fix it.
