The 357 Sig is an easy round to load, once you figure out the few little quirks. I've loaded well over 10,000 rounds of it so far.
As snuffy pointed out, use only flat point or hollow point bullets, as the overall length of this round is pretty short. The advice to use .355" or .356" diameter bullets is also correct, as the 357 in the round's name is only that, a name, since it was designed to compete head to head with the .357 Magnum 125 grain loading, only in a pistol configuration. That's why there is no "." in front of the "357" in the name. I load a lot of Berry's 124 grain plated flatpoints and plated hollowpoints in this round, but you'll also find that the Speer 124 grain Gold Dot specifically made for the 357 Sig is also a very good bullet, as well as the Hornady 124 grain XTP. Also use a powder that fills the case and is slightly compressed, as this helps with preventing bullet setback. I prefer Accurate Arms #9 for my loads, since it's proven to be very, very accurate. You can use any standard small pistol primer for this round.
You may run into a problem with the Dillon dies, since they are belled at the mouth of the die to facilitate use in a progressive press. Mine wouldn't size the case far enough to take the swell out just ahead of the web, but I solved this by running the brass through my Magma Case Sizemaster jr., which pushes the entire case all the way through the die and sizes the whole length of it, including the rim. I already had a 10mm die for the Magma, and the 357 Sig is based on a 10mm size case. Of course it won't size the neck, but I do that in a my Dillon 357 Carbide size die when I punch out the primers.
Speaking of depriming, you'll find that some Speer brass has a very small primer flash hole. So small in fact that the Dillon decapping pin won't go through it. I'm told this brass is made for Speer by another vendor, but it's good brass. You can tell the difference by looking at the base of the case. Some Speer brass is totally flat appearing and other Speer brass has a rounded appearance. I drill all my flash holes the same size, once I use a universal decapping tool to punch out the original primers. More work, but I've devised a way to do it really fast, which I won't go into now.
The round is designed to headspace on the case mouth, but I've found that there are two actual datum points for headspacing. One is the case mouth and the other is the shoulder. Some dies won't set the shoulder back far enough for the round to chamber. I had one die/shellholder combination that wouldn't allow this to happen, so I milled off about .008" from the top of the shellholder, to allow the case to be pushed far enough into the die to set the shoulder back to where it needed to be. It was an easy fix and I've helped others with this same problem. There is enough tolerance stack between the dies and shellholders to sometimes cause this problem. I'm not sure how this is going to factor in on your Dillon press, since I don't use one.
Another area that might give you a problem is the seating of bullets. Because of the short neck, you need as much neck tension on the bullet as you can get. I choose not to bell the case mouths, since I want all the neck tension there is. I chamfer all my case mouths, which only has to be done one time. This allows bullets to seat into the tight neck without shaving the jacket, including the plated bullets, which are considerably softer. Once I did the chamfering, all seating problems disappeared. I've also devised a process for chamfering whole batches of cases really quickly and it only has to be done one time.
I also crimp, using the Dillon crimping die, but I canellure all my bullets and crimp into the canellure, which helps to keep the bullet from setting back during the feeding cycle.
I've found that Federal nickel brass has a tendency to be long. Some of it is so long that even once fired brass won't chamber. I've bought a file trim die for this round, but what a pain that is! Since the brass is now pretty plentiful, I would just recycle any brass that is too long to reload. Just watch the length of the cases and when you start to get some that are longer than SAAMI maximum, look for some newer brass.
I have two pistols for the 357 Sig, a Springfield XD and an EAA Witness. Both are exceedingly accurate with this round, but the XD prefers 124 grain bullets and the Witness prefers 115 grain bullets for the tightest groups. Both will shoot ragged one hole ten shot groups from the bench @ 15 yards, and keep all ten rounds within about 2" at 25 yards, if I do my part. I've saved some targets that even amaze me. This is one accurate round.
If you have any questions, be sure to ask.
Hope this helps.
Fred