Sway bar links won't affect your vehicles ability to turn, but will heavily increase the amount of "sway" often referred to as body-roll you feel while cornering. Sway bar links and the sway bar itself are attached via mounts to the vehicle frame or sub frame, and from there the links are attached to the control arms of your vehicles suspension. As for an amateur DIY mechanic, the setup all depends on how comfortable you feel with fixing your own vehicle, and knowing what's involved (in case you do make a mistake).
However, most sway links I've come across haven't been too difficult, although installing the new parts often takes a bit of grunt work and muscle to get the bushings and bolts through the sway bar and into the control arm. When in doubt, take it to your local garage and have a certified technician perform the work, and alleviate yourself one big headache. Best of luck to you!
4 Years as an ASE certified Mechanic