What's a common way people go about dealing with large amounts of stress, and the loss of loved ones (parents, siblings etc)?

Accepting grief. Grief is a natural reaction to loss and involves various stages: denial, bargaining, anger, depression and finally acceptance. We all go through stress when we experience loss. It all takes time so we need to be patient with ourselves and the process. Accept the pain and hurt. Allow yourself to experience the grief because you need to go through it to get through it.
Talking to someone. I think it is a good idea to speak to a therapist, attend group therapy or enlist the help of your clergy.