The snare traps can be bought (224G) at Sophia's shop and be set anywhere, but only really have a decent chance to catch something if you fulfil 3 requirements. These can be seen once you have set the trap and selected to add bait.
The first icon is for how strong the bait is for attracting animals.
The second is if the trap is in the right environment (typically anywhere on the map coloured green and with tree icons). If this is still showing red, it may be too close to an occupied area, try moving it a little way further from the buildings.
Thirdly there is the icon for how far the nearest other trap is, it seems to be about 6 fence lengths apart at the closest distance.
More than 1 too close will lower your chances further...

There are 2 types of bait, strong and weak. The stronger ones seem to have a sure chance of attracting an animal and weak seems to be a 50/50 chance.

Strong

• Carrots - Always catches boars.
  
• Blue berries - Deers and stags
  
• Raspberries - Deers and stags
  
• Eggs - Foxes

Weak

• Onions -
  
• Sliced bread -
  
• Ham slice -
  
• Sandwich -
  
• Apples -

Guns
All guns are available at Sophia's shop, She also stocks 50 ammunition per day (12G). There doesn't seem to be a difference in power but instead each upgrade increases accuracy and magazine size.

• Rifle - 1 shot, 1344G (There is one on the table in Kasimir's barn).
  
• German Rifle - 5 shots, 5600G.
  
• American Rifle - 15 shots, 11200G.

Explosives
Sophia also sells a few slightly more devastating items for those who are bored of fishing. These can also be found in varying amounts around the map.

• German Stick Grenade - Small explosion, 448G (5 per day).
  
• Russian Grenade - Larger explosion, 448G (5 per day).
  
• Mine - Once set, it can be triggered by firing your rifle at it or when something walks on top of it. 168G (3 per day).
  
• Bomb - Kasimir can combine 3 mines, a stick grenade, a russian grenade and a peice of scrap into a bomb. Once set, it has a timer to detonation. Possibly overkill for sorting the fox problem but works a treat on clearing old tanks and ruins to get 10 scrap or 5 planks and 5 asbestos respectively.

Melee?
There are other ways of ending the lives of critters but they are much more tricky to use. Any sharp tool can do the job but it will require you to get up close and personal with your prey. The pitchfork definitely works, so maybe be careful cleaning out your stables... poor horsey.

Other...
The final offensive item is a bucket of manure. No good for hunting, but great for upsetting the locals. Be warned, they most likely will retaliate.

There are several wild animals Kasimir can hunt in Farmer's Life and he can butcher them all using the blood soaked table in the barn.

Birds
There are plenty of crows and herons flying about and sitting on roofs, they go down in a single shot and can be butchered for 2-4 poultry (12mins).

Game
These are docile creature that will turn and run when they notice Kasimir. He can crouch to reduce likelihood of being detected and it also helps to keep a steady aim on his target. Or he can bait a snare trap to catch them over time.

• Boars - Takes 2 head shots or about 6/7 body shots to kill. Produces Boar hide, Boar trophy and 7 meat when butchered (39mins).
  
• Deer - 1 head shot or about 3 body shots. Produces Deer hide, Deer trophy and 7 meat when butchered (39mins).
  
• Stag - 1 head shot or about 4/5 body shots. Produces Deer hide, Stag trophy and 11 meat when butchered.
  
• Fox - 1 head shot or 2 body shots. Produces Fox hide, fox trophy and 4 meat when butchered (18mins).
  

Albino give the same as their regular counterpart except white hide and trophy.

Bears
Bears only spawn in 3 set locations (West of Ann's farm, South East of Kasimir's farm and around the abandoned church) after a certain amount of time. If they detect Kasimir, they will attack. They hit hard so be careful and crouch to avoid detection. The bears can attack other nearby villagers and animals, the villagers will blame Kasimir and loose affection/ not talk to him just as if he had beat them up.
To take down one of these natural tanks take 7 head shots or 15 body shots. They will produce a Bear hide, Bear trophy and 26 meat when butchered (1h30).

Fishing
There is a fantastic quick guide to fishing by Kruzenschtern.
https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=2928046135