The legality issue is something that needs to be closely examined. Just because you think you know what the law is, does not mean that you really do. Please don't take this the wrong way, gun laws are, unfortunately, often very confusing and difficult to understand and many people have unintentionally become criminals; "ignorance of the law is no excuse".
Something that could potentially be very important is exactly how old you are. "Underage" really tells us nothing more than that you're under 21. According to federal law, you may posess a handgun or buy a handgun from a private individual at 18 years of age but you may not buy a handgun or handgun ammunition from a federally licensed dealer until you're 21. In most (but not all) states, you may transport a lawfully posessed handgun unloaded and locked in a case at age 18, but carrying a loaded gun for self-defense is a stickier situation. In order to carry a loaded gun, most states require you to have a license or permit of some type. These licenses are usually not issued to people under the age of 21 although there are certain exceptions to this (Indiana will issue a license at age 18 and Alaska, Vermont, Arizona, and Wyoming do not require a license). While there may be exception for handgun hunting, simply being in posession of both a hunting tag and a loaded gun usually isn't enough to allow you to carry said gun legally.
As to a choice of handgun, unfortunately there isn't any one single gun that can do everything (or at least not do everything well). As an all-around gun, a .357 Magnum revolver with a 4" barrel is about as good a choice as you can make. The .357 Magnum is an excellent self-defense cartridge and a 4" barrel is long enough to be a practical field gun but short enough to not be overly cumbersome for other purposes. That being said, a .357 Magnum is not an ideal hunting revolver. The cartridge, while a good one, is really only an acceptable choice for game up to the size of a deer (and it's considered marginal for deer-sized game). Likewise, while a 4" barrel is useable for hunting (though it may be to short to be legal for that purpose depending on your state's regulations), a longer barrel is more preferable because it will give you both higher velocity and a longer sight radius while the extra weight will help to dampen recoil.
A larger .41 or .44 Magnum revolver with a 6" or longer barrel would be better suited as a dedicated hunting gun. However, such revolvers are much larger, heavier, and more cumbersome than most .357 Magnums and are too big to be a practical concealment gun for most people. Likewise, the recoil of a .41 or .44 Magnum is quite heavy and is usually only well controlled by an experienced handgunner. Finally, while .357 Magnum ammo isn't cheap, .44 and particularly .41 Magnum ammo is rather expensive and most people who shoot those cartridges in any quantity handload their ammunition.
For a good all-around revolver in .357 Magnum, a S&W Model 19, 66, 586, 686, or 620 would be a good choice as would a Ruger GP100 or Security Six. For a dedicated hunting revolver, a S&W M57, 29, or 629 would be good choices as would a Ruger Redhawk or Super Redhawk (there are single-action revolvers that would be good hunting guns as well, but you said you wanted double-action).