I don't know how much .44 Magnum shooting I could do, because I've fired a grand total of one .44 Magnum round in my life. But as I recall, it recoiled a lot harder than my .357 Magnum guns!
For target shooting, the .357 Magnum is going to be less expensive per round. Of course you can probably shoot more of them in one session, which might offset that expense. The same applies to the Special rounds that can be fired in the appropriate Magnum guns. The .44 Magnum makes a big boom, but unless you're involved in a shooting sport that calls for a cartridge of that size and power, the .357 is probably the better alternative.
For self-defense at home, the .44 Magnum is more powerful than the .357 Magnum. But the .357 has a well-earned reputation for noise and flash, particularly indoors, and I can only imagine the noise and flash of a .44 fired indoors. I am reverting to .45 ACP automatics for first-line home defense handguns - quieter, less flash, higher capacity - and putting my GP100 in the second line.
If you don't mind carrying openly and you spend time in the woods, a .44 Magnum revolver would be a fine carry gun.