Mary, it's up to you, what you feel is the best to do.
When my dog died, I burried him in our garden, in to his favourite spot, under few little trees, he liked it there, he could watch the world, but the world didn't see him.
I liked the idea of having a grave, so I was able to go and spend time there when I needed. I put some flowers there, and at first, I kept sitting there everyday, because I missed him so, and I felt lonely. I though that while I sit there, I'm with him, he's beside me and watching the birds with me. This went on for months, and when winter came, I put candles there and at x-mas, I decorated it a bit, it felt good because my dog loved x-mas anyway.
Sometimes now it's breaking my heart because I can't visit his grave anymore (because I live sooo far) but I know my parents take care of it.

Sorry to go on and babble so much, I guess I needed to vent
I hope you find a solution that is the right one for you. I know you will!