Hey Ali
Sorry about your granddad. I've lost all my grandparents now. My dad's parents were more like my mom and dad than my grandparents. Grandpa has been gone 23 years and grandma for only 2 now. I miss them both terribly! You'll always miss him but with time, it will get easier. The 1st holidays are tough without them. I believe you have the right of it to put the tree up and go on with christmas. The best way to honor someone's memory is to celebrate their life and do the things they loved best.
You and the family could do something special to celebrate as well. Something like....getting some air dry clay and making a special ornament for him, set up a spot on the mantle or under the tree even....like a "altar"...only word I can think of to describe it. Put a couple of candles (if it's on the mantle,use electric candles or a small strand of xmas lights under the tree) a photo of him, a bit of holly or mistletoe, a favorite object he had. Anything really that's meaningful to all of you and him. Just a few ideas for you and the family and maybe that'll spark some idea in you as to what to do.
My grandpa loved christmas himself and I always think about him more at this time of year. I also try to honor his memory in some small way every year too. Oh....you could, maybe not now, but over time, write down your memories of him, things you did together. "Interview" your nanan about their life together, special times, hard times, the day they were married, etc. Write it all down then put together a scrapbook with the stories and pictures in it. Not only would it be a keepsake but something you could share with your own children one day.
I'll be thinking of you and your family and wishing you easier days ahead.
Hugs,
Karen