When it comes to kit, if you're using interchangeable lenses, pop a macro lens on your camera while compact camera users can just switch over to macro mode. You'll also need a tripod to help keep shots steady as shutter speeds can be slow when working in between Christmas tree branches.
Before you start taking your shots check your white balance as you can end up with shots that have an orange tint. Try using the Tungsten setting or shoot in RAW and adjust your shots later in post production. Put your camera's self-timer on while you're in the menu system so there's a few seconds for any shake to stop before the photo is taken.
Often people try to take a photo of a section of their tree rather than focusing on one ornament. This means your photo doesn't have any focus and the viewer's eyes will end up darting around your image. To stop this, get in close and fill the frame with just one of your tree ornaments. If your tree's so full you can't get away from groups of baubles then throw the background out of focus. You can get some interesting bokeh from the Christmas lights but you'll need a fairly small f number to do this. The lights can create interesting patterns and leave subtle spots of colour on the ornaments adding another level of interest to your image in the process. Try to stand so your reflection doesn't bounceback off the ornaments though.
Decorations that are colourful or have plenty of interesting detail are a good choice but something more simple can work well when you're playing with bokeh in the background of your shots.
While you have your macro lens out take your eyes away from the tree and look for table decorations, ornaments on the mantelpiece or even tins and bowls of colourful chocolates/sweets to photograph.