Soldering is great, but the fumes, less so. The link between solder fumes and asthma is clear. More fumes, more asthma. Breathing (and eating) lead is also not recommended. The newer lead-free solder compounds may be safer in some ways, but also contain other likely harmful substances that we are better off without.
Soldering can produce smoke with particulate matter, volatile organics, and other noxious gases. A good system should both filter out the particulate matter, and then exhaust the remainder to the outside.
This system uses a shop vac with hepa filter to provide local suction and manage particulate matter. Exhaust from the shop vac is then fed into an exhaust fan that is ducted to the outside.
Here's a video of the system in operation:
The core of the system is a good exhaust fan. This fan handles my solder exhaust, but can also improve the ventilation of my work room in general. If you are trying to exhaust the entire room, you should probably calculate the "air changes per hour". A good number for ventilation is 8-20 air changes per hour. You can do the math, but for 8 air changes per hour a good rule of thumb is 1 CFM per square foot. At a minimum, the CFM for the exhaust fan needs to be higher than the CFM for the shop vac.
I use this one Vivosun Exhaust Fan. It seems to be used primarily by indoor marijuana cultivators so can move lots of air, relatively quietly.
I use a combination of PVC pipe and metal ductwork for the exhaust unit. This let me create a well sealed hole through the wall as well as using a cleanout fitting to completely seal the exhaust vent when not in use.
The particulate matter filtration and suction boost is handled by a shop vac with HEPA filter. I use this 4.25 HP Rigid Shop Vac from Home Depot, but you can use whichever one you already have. It's a bit noisy, but works well. The main suction hose attaches to the PVC suction arm. A second hose from the shop vac exhaust rests inside the intake vent for the exhaust fan.
The desktop suction arm is made from PVC. In order to attach the shop vac hose to the PVC, I had to dilate the last pipe a bit. I used the same technique for adapting my shop vac to serve as a dust collector for my miter saw.
I created a cone shaped mold out of cement to dilate the PVC pipe. Here's a link with a video to show the process.
Finally, I stained the PVC black to match the decor of my work shop.