I found one of the easiest ways to simplify the game without hacking or removing many sources of Fun is to lower the population cap. I found that once my fortress grew to 100 dwarves or beyond, I couldn't figure out how to manage all of them and keep them working productive jobs. So instead they all just sat around throwing parties and pestering my productive dwarves. Until you get enough experience to handle a fortress that size, try lowering the cap to 40 or 50 dwarves.
To do this in v0.31.08, edit data/init/d_init.txt and change the value of this line:
[POPULATION_CAP:200]
Note: your actual population may still go above the cap you set, you just stop getting new immigrant waves after you reach or surpass that cap. And apparently births don't respect caps. Also I think (but I haven't tested) that you can change this mid-game if you save and quit first.
Assuming you want to capture animals (groundhogs, horses, bears, dreaded Rhesus Macauqes) as opposed to vermin.
Build a cage trap where the animals roam (usually next to an outdoor food pile works well) with build, Traps, cage. You will need a mechanism and a cage. Mechanics skill is used to place the trap.
You should probably designate an animal stockpile (p, a) where the animal will be taken after it's caught.
After this, just wait for the animal to be caught.
Note: the capture live land animal option at the kennels catches vermin not animals.
Hint: placing your cage traps randomly can take too long to capture anything.
One way to increase the chances of an animal walking into your traps is to channel one long, 1-tile wide, dry moat(d,h) and remove all ramps(d,z) except one in the middle of the moat. Then you surround the single ramp with traps.
Once you've captured an animal, even if you're going to butcher it you probably want to get it trained at least a little, to avoid it breaking free on the way to the butcher shop. To train, make sure you have at least one dwarf with 'animal training' labor enabled, then find the animal in the status tab (z) and designate it for training. Butchering is also handled through the same status menu.
Best Answer
According to the wiki, while you cannot force a dwarf to adopt an animal, the dwarf must be wandering nearby the animal. In my experience, since caging cats will prevent their adoption, the animal must not be caged. So the most likely step is to let the vermin out of the cage that they are in. I have not applied this specifically to vermin, but I believe that it still holds. I do not remember whether or not loose tame vermin congregate in the dining hall like other stray animals. To be safe you might want to release them from a cage in the dining hall.
You still cannot force adoption of anything other than working animals, but this should make it possible. Be sure you want to do it though, as a pet's death can cause an unhappy thought and vermin die rather quickly (bats for example only live for 2 or 3 years).
So to reiterate the steps to having a pet vermin:
you must have captured it in an animal trap
tamed it
marked it for pet availability
let it wander loose so dwarves can meet it and decide that it is the most adorable thing ever. ("Who's the cutest little purring maggot ever? you are! yes you!")
This last step is the one I believe that you are missing.
Hopefully this is of some use.