Budget and intended use? Do you want new-production only, with easily accessible customer service, or would you look at older guns that, to varying degrees, could be difficult to service, should the need arise? There's a vast range of options.
The S&W Model 686 (and 586) and Ruger GP100 are the standard suggestions for new sub-$1,000 .357 Mag. revolvers. Neither do much for me, but they are very solid guns.
S&W's flagship revolver has long been the N-frame Model 27 (and pre-Model 27 and Registered Magnum). If you're okay with the unfortunate internal lock (present on almost all S&W revolvers since 2000), the absence of pinned barrels and recessed cylinder bores (both eliminated ca. 1982), MIM parts (present on all S&Ws since 1997-98), two-piece barrels (not sure of first year), and bluing that doesn't compare with the S&W wheelguns of yesteryear, then you can buy a new Model 27 "Classic" (27-9) for not much more than a 686. The Model 27-2 is the last version to have none of the aforementioned changes that most S&W enthusiasts dislike at least to some extent. In general, the further back you go, the more expensive the gun will be, all else being equal (though there are exceptions, with the rare Model 27-1 being more valuable than an equivalent 27 "no dash," for example). A pristine pre-Model 27 can run several thousand dollars these days, depending upon barrel length, and a high-condition Registered Magnum will easily run into five figures in even the most common barrel lengths. The Model 28, made from 1954 to 1986, is the same as a Model 27 from the same production year but with a few features and options omitted to keep production costs lower (no checkered top strap and barrel rib, matte blue finish rather than high-polish "Bright Blue" finish, no nickel finish option, fewer barrel length options).
If you want the finest DA .357 Magnum revolvers ever made, then you're looking at Ratzeburg-made Korth Sports and Combats and Mulhouse-made Manurhin MR73s, with a well-preserved example of any of them coming in at several times the cost of a new S&W or Ruger. If you want the finest DA revolver currently made, you're looking at the aesthetically challenged Lollar-made Korths (imported by Nighthawk). (Not really counting Janz here, since production is so low.)
If you want the finest SA .357 Mag. revolver ever made, look at Freedom Arms. The USFA revolvers made after they stopped using Uberti parts are also very, very nice (and the ones with Uberti parts are still nice). (On the much cheaper end, people love their SA Rugers as well, but I have little experience with them.)
There's also the K-frame S&W Model 19 and 66 (and others), but these are smaller and lighter revolvers, just like your Model 10. Many of them will not hold up very well to a steady diet of full-power .357 Mag. ammo. In recent-production guns, S&W eliminated the flat gas-ring clearance cut on the bottom of the barrel at the forcing cone, thus strengthening the barrels and increasing longevity. You'll have to determine whether they are too light to make pleasant shooters.
Plenty of Colts you could look at as well. The old Pythons are as iconic as .357 Mag. revolvers come. They aren't all of the same quality, though. Colt's quality had really declined by the late 1970s, and it bottomed out in the second half of the 1980s. Some of the 1990s guns I've seen (and owned) aren't much better. I don't have any experience with the new Python. It has some significant technical improvements over the old version, but I've read more than a few complaints of serious quality control issues over the last year. The only other Colt .357 Mag. revolvers I like at all are the original Trooper (1953-69) and the Colt 3-5-7 (1953-61), which was just a more refined Trooper (or, in many respects, a less refined Python). (Well, I also love the Shooting Master, but only 500 were made in .357 Mag., and they aren't cheap.) Other more common old Colts in .357 Mag. are the Trooper Mark III, the Trooper Mark V, and the King Cobra. The new King Cobra is a small-frame revolver of a different design.
Dan Wesson revolvers have a smaller but ardent following. I gather that quality has varied at different times, with guns made in the late 1980s and 1990s being viewed less favorably than the earlier guns. Current-production DWs seem to draw very positive reviews overall.
There are other cheaper options from Taurus, Rossi, Charter Arms, and RIA/Armscor, but the 686/586 and GP100 would be my personal floor in a search for a quality .357 Mag. wheelgun.
Plenty of other brands and guns that I haven't listed as well.
