Nice revolver, in a great caliber! Your grandfather had good taste in firearms,
There are two basic loads for what you want to do: a mild target load for familiarity and practice and a stouter load that you will drake your life on (hopefully not, but you get the idea).
If you are not (yet) a reloader, then any bulk-pack non +P .38 Spl ammo will suffice. As was stated the loads shoot to different points of aim sometimes, so try to find one that best approximates the point of aim of your chosen defense load and then you won’t have to fiddle with sights or change your sight picture much. This may require a bit of load experimentation that requires buying different brands and bullet weights, which isn’t easy when people are buying any gun they see and boxes of ammo to go with it.
As for the Hornady Critical Defense load you mention, I’d shoot the +P version for Home Defense (HD) . (I use it in my lightweight .38’s while the Diamondback is a mid-size revolver.) The Diamondback is stout enough for +P use in such a role and that ammo has a bit more oomph should you require its services.
Federal HST is also great stuff, as is Speer Gold dot, Winchester Ranger Bonded and Remington Golden Sabre. They’ll all work well in a HD role if you put bullets where they need to be to stop a threat.
If you’re considering stating to reload, the .38 Spl is a fantastic first cartridge to try out. Many of us here cut our reloading teeth on it, as bullets, primers, powder, brass, dies
and load data is abundant. I’ve loaded countless thousands of .38 Spl rounds (500 alone the weekend before last) and I find it to be very forgiving and fun to shoot my own stuff. (I haven’t bought a box of factory ammo in .38 in probably 15 years now)
A Lee challenger kit with dies in .38/.357, a pound of Unique, 500 158 gr LSWC and 500 small pistol primers, a bunch of once fired brass bought off of here or Gun Broker and you’re set for lots of loads and reloads without spending a fortune.
Welcome to THR, and stay safe!