yes everybody has different beliefs. I read somewhere that your dogs soul will stay 'at the other side' of Rainbow Bridge if thats what the their owner believes.
People believe in a variety of gods (depending on their religion) or angels etc.
Some people chose to tend or visit a grave reguarly, everyone's different.
I suffer from depression and there was a time I was planning that my life would end when Inca's did as I felt she was all I had.. but then in a period I was feeling more positive I decided I wanted to train another dog up.
When she was very ill earlier in the year I had similar thoughts and that JJ was young enough to settle in another home if I wasn't here, but then I discovered these stories from people all over the world on different sites (not just that one, I found that the clearest description to understand) who had stories of other dogs that came into their lives and would do things exactly the same way a past dog did.
My family believe in cremation anyway and that the process separates the spirit from its earth body, so it wasn't such a big leap of faith for me. As dogs (or other pets) don't live as long as humans they can 'come back' in the same lifetime to be with the same person again if they want to. I found this a great comfort and thought others may do too that are going through the same.
Whether they decide to click the link or google others is up to them.
I found this page interesting too
Reincarnation Reunions