It depends on what the game does and starts when you take your shot.
If the game drops:
- Do NOT move. Sit and watch it. Wait at least 15 minutes. Often game will drop and freeze, even when injured, then stand up minutes later and stroll off.
- If it stands, shoot it. If you miss again, follow steps in the next part.
If the game runs:
- Mark where the game was in your mind. Get rough distance from
two easily visible points.
- Go to where the games was and inspect for tracks.
- If you find no blood in the first 100 yards it is incredibly unlikely that you hit your target (unless you're hunting elk, sometimes their fur mats the blood and leaves no blood to follow)
- From where you took your shot, imagine a straight line 10 ft out. Inspect that line for blood, fur, and parts. You can get an idea of how good your shot was. If you see bright pink, foamy blood, you got lungs and its an easier track. Lots and lots of blood is usually a heart shot. Rub it between your fingers and smell. Gut shots usually smell really foul. Gut shots can take hours to find.
- Stalk the animal. You're now in a hunt again. Act like you're in a hunt. Don't rush.
- The animal is (hopefully) injured and scared. It will likely hole up. If you come to dense thicket or brush, send someone to the side (careful to plan fire lanes) in case you flush it out.
- If you see the animal, and you are SURE it went into a thicket, don't flush it. Wait 15 to 30 minutes. Once the adrenaline wears off it is much less likely to stand up and run with that injury.
- Track prints, presumably if you are hunting you have practiced tracking. It's hard to boil tracking down into an answer but: Note size of prints, note direction, and don't expect to find every print. When you have a print range in a five foot diameter to try and find the next.
- Track blood. Blood does something you won't often otherwise see in nature that makes it easier to follow. In small drops it tends to make near perfect circles. Even if you cannot see the color, the circles tend to stand out. Even being colorblind I have spotted blood this way.
- Walk VERY carefully. You don't want to obscure sign.
- At night, fresh blood shines well, and you can easily spot the shine in a flashlight beam.
- If you can get a dog. Even a dog not trained to track will usually tag onto injured game. We once found a gutshot dear which ran five miles with almost no sign only because we borrowed a dog from someone. Even though the dog had no training, it located the deer by smell and instinct.
To keep the meat you want to find it fast. But if you don't find it in the first 300 yards or so your priority really changes to just making sure that it's not lying suffering somewhere. Take your time, be patient. Plan ahead before you hunt. When you hunt, you need to plan to have the next 10 hours clear. You don't want to have to stop tracking and leave a suffering animal because you made afternoon plans.
The clean kill is kind of a myth. The heart isn't a particularly fast death, just a more certain one. I've seen game run 300 yards with no heart, and stand up with no brain. Who's to say what they feel during that time. The heart is a target because it's in a cluster of vital organs and if you miss a couple of inches in either direction, you probably still have a kill. However if it's a lung shot you could still be in for miles of tracking. If you miss a couple more inches and nick a leg then you may never catch up with the animal (which may even live). From a strict biological standpoint, hunting is not a cleaner death. From a touchy-feely one, perhaps living free makes up for that. But don't go hunting planning for the "clean merciful kill".