Ammunition qualities: 1. Inexpensive, 2. Universally available, 3. Reliable, 4. Get a cartridge made for many years so there is no chance it will be discontinued due to market forces. 5. You want an accurate round, pleasant to shoot, that does not dish out too much obnoxious recoil. 6. Most of all you want a FUN GUN.
Also: A good round is one useful for many puposes.
Your gun should have the same qualities.
I am an auto fan but that said I think beginners are best served by revolvers, and I still use a revolver too. With revolvers you have a better selection of ammo types in general.
In a handgun, such a jack of all trades, master of none is the .357 Magnum, versus, say, the .44 Magnum which is more a hunting round than anything else. The .357 will also shoot all .38 Specials, .38 Long or Short Colt rounds and some .357s will digest the .38 S & W which is most practical. The .357 at short range will also take deer but it is marginal for that purpose. The .357 is not the best target round, but for informal target shooting is fun and fine.
If you insist on a deer round in a handgun there are several that qualify but all of them dish out the recoil. The .357 top loads are not too obnoxious, and represent the most that the average Joe wants to put up with.
Stay away from the very rare or specialized rounds, they are less fun, cost more money and are harder to find. An example of these is the 10 mm Auto, which was touted as a defense round but kicks too much. This comment does not apply to the specialized self defense numbers but don't you want a "fun gun" too?
For economy the king of the hill is the .22 rimfire, but it is not a good defense pick. On the other hand, in the right gun as a small game and target number it is past master.
Great guns are made by S & W in revolvers, and Ruger or Springfield in autos.
Big kicking guns like the .44 Magnum are best for hunters. The .44 Magnum will digest the mild-mannered .44 Special. But the .357 is better because you can shoot MILD loads that STILL have a flat trajectory for fine work out past 50 yards. As an example of this, with my .45 Auto I once missed three shots on a cottontail at 100 yards, the bullets going OVER or UNDER this rabbit. The reason is the .45's slow velocity and curved trajectory. Had I used a .357 or even a .22 Long Rifle that bunny would have been cooked.
A good thing about the .357 or .44 Magnum is the availability of nice, light lever action RIFLES in these rounds. These will greatly extend the range and augment the stopping power of these rounds. If you dig the old cowboy concept of two guns needing only one round, to simplify supply, these rounds might be for you.
The .45 ACP is the best defense gun, but I can sometimes watch the bullets in flight, they are slow at about 450 to 600 m.p.h. or so. Do the math and you will know. 800 fps x 3600 (seconds in an hour) over 5280 = 545 mph. Of course, you never dare let yourself be hit by one.
A great round for the beginner if you can get a quality gun is the .22 Magnum. This rimfire round actually has a power level that is respectible and while marginal for defense would not be a bad choice for the beginner. As a game gun it is quite good on stuff smaller than deer.
If I had to do it all over again my choices would be, in this order.
1. The .357
2. The .22 Magnum convertible such as Ruger makes with an extra .22 Long Rifle cylinder.
3. A .44 Magnum
4. A .45 ACP
5. A .45 Long Colt (I have a soft spot for this famous but still popular old number).
Also, think on if you desire a single or double action revolver. The old cowboy guns are a reliable rugged design but the long fall of the hammer takes more practice on the target range.
Chris