There may be anger at the vet for putting the pet to sleep, anger at other pets that are still alive, just for being alive or even anger at the deceased pet for needing to be put to sleep. You may lash out at other family members for, as you perceive it, talking you into putting the pet to sleep. You could even feel anger at yourself for making the decision to put the pet to sleep. Just remember, anger stems from hurt. When we as humans are hurt so deeply, we tend to express our pain as anger.
Guilt is another natural part of grieving.
Everyone feels guilt about needing to put a pet to sleep. Guilt often goes hand in hand with denial, in the thinking that if you had waited longer maybe your pet would have gotten better. Remorse is one of the longest lasting stages of grief. Remorse can last anywhere from days to years. The loneliness stage of grief can occur simultaneously with remorse.