Of your 2 initial choices, the 3 inch model 686 is a superb choice. If you can, get the 686 Plus version because it's a 7 shooter as opposed to a 6 shooter.
For home defense ammunition, go with a 38 spl. +P premium self defense round. In a Home Defense situation there really isn't any need for a 357 Magnum and the 38 spl. will be quiet enough that there will be a lot less risk of permanent damage to your hearing. Out of a 3 inch barrel a 357 Magnum will be loud enough to cause permanent hearing loss or even total loss of your hearing. Since you won't be shooting thru any steel car bodies, there really isn't any need for the much more powerful Magnum. In addition there will be much less risk of over penetration with the 38 spl.
For range use, standard 38 spl. will be just fine and close enough to the 38 +P that any deviation in sighting will be well within 1 inch at 40 feet or less. Recoil will also be quite similar to the 38 +P so that training with the less powerful 38 spl. will have very little effect on your ability with the slightly more powerful 38 +P.
As for those who have suggested a shotgun or rifle for Home Defense, they haven't taken into account the layout of YOUR home. I live in a small home with narrow hallways and there just isn't room in many areas that would allow me to swing a long gun. If your in a similar circumstance, a handgun is a much better choice. In addition, some shotgun calibers can be every bit as loud as a 357 Magnum, so damage to your hearing is a real potential. Finally, it's a lot more difficult to engage multiple targets at close range with a long gun than it is with a handgun because you have to fight all that inertia. Quite simply many times a handgun you have practiced with is a much better choice for close range defense than any long gun.
As for the price, 769 dollars for a 3 inch 686 is fairly good. I can do about 40 dollars better in my area but I live in a large metropolitan area with large high volume dealers. Bottomline, that's not a bad price for a brand new gun.
Tips, pay for training from a professional. Your level of skill will grow much faster with professional training. You'll also want to do a lot of training initially, plan on once a week for a minimum of 8 weeks because it will take that long to learn how to master shooting well rapidly with the Double action trigger. BTW, shooting well means hitting a 5 inch target ay 30 feet while shooting at a 1/2 second interval time. At first that will seem impossible but after 8 weeks of steady training you'll find it pretty easy.
Note, many ranges will ban shooting this rapidly. However it's been my experience that if you frequent one range frequently and get to know the people, they will waive the firing rate rules when the range isn't in heavy use. Many times dinner hour during the week are slack times at shooting ranges so you might want to make it a habit to eat late once a week. I have also find it very benefical to ask for permission before engaging in any rapid fire practice. Many times if the range isn't busy they will say yes and have a Range Officer watch how you do. Get to know that Range Officer and many times he'll point out what you are doing right and where you could do better.
Also, remember you are practicing to defend yourself, NOT target shooting. While a 5 inch group may not sound that good, put a tape measure to your head. If you can consistently hit a 5 inch target at 30 feet, you can certainly shoot well enough to defend yourself. This means don't cheat and spend your time shooting in single action, practice how you may have to use it and master shooting in double action.
Final note, purchase some Azoom snap caps for the new gun. They will protect the firing mechanism when you're doing dry fire practice at home to strengthen your trigger finger and refine your trigger stroke. Also, watch the sights when you practicing with dry fire, the goal is to pull the trigger in one smooth rapid stroke without causing the gun to "shake". Start by stroking the trigger slowly in one smooth motion and build up speed as you get smoother.
PS; keep in mind that this isn't the Movie of the Week. If you are facing an armed Home Invasion your only obligation is to protect yourself and your family. This means that you Identify the target, Confirm they are armed, Aim and Fire. You don't yell "stop or I'll shoot" like we have all seen in the movies or on TV, you shoot them down like you would a rabid Pit Bull. After they are down and no longer trying to do you harm, you may then disarm them and call 911 and provide any medical care you think is appropriate until help arrives.
If you're facing a Home invader who is not armed, you have a bit more leeway in how you react. Just be aware that they may be concealing the fact that they are armed. If there are any signs of resistance or aggression, it may be best to treat them as if they are armed. This is where Castle Doctrine comes into play, in most states you have the right to defnd your homestead with lethal force, even against an unarmed intruder. However, you do want to familiarize yourself with your states laws in this regard.
I personally think that it's bad form to shoot a drunk who has kicked in the wrong door. I also think it's bad form to shoot a teenager looking for an easy score with the thought that nobody is home. If some unarmed idiot were to kick in my door he would be presented with 3 choices, given at gunpoint. 1) Run like hell. This is assuming that he runs as soon as he sees me and I don't have time to say anything. 2) Fully comply with every order I give. This means that he has to strip down to his underpants so I can see he is unarmed and call 911 on himself. 3) Get shot for showing resistance or aggression. I'm an average mid 50's male and have no illusions that I can go toe to toe with someone younger and fitter.