All solder will release fumes, the majority of which is flux contained within the solder which helps to clean the soldering points. Lead is dangerous, sure, but if you take proper precautions such as washing your hands after soldering you won't have much to worry about. If lead really bothers you, you can get lead-free solder, but be aware that it is much more difficult to work with since the melting point is much higher. If you go the lead-free route, you will need to ensure that you have a very good soldering station (preferably with temperature control) that can reach at least 425F (220C), though 450F (232C) would be preferable. This is in contrast to leaded solder which melts at 361F (183C) for 63/37 or 370F (188C) for 60/40.
The other issue with flux fumes cannot really be solved unless you find a specialized solder, but there are none that I know of and if they do exist they will likely be very expensive. It's far more beneficial to just get a fume extractor which will suck up the fumes and filter the air (I use a $20 dehumidifier with charcoal filters... works great!). I also recommend using either a dedicated workspace for soldering, or making sure that the space you are using is properly covered, e.g. plastic tablecloths will work well.
