I would just spend some time looking at the customs here, searching youtube, and looking at Pinterest. If I were to try to build with wood though I would definitely go with a burnt wood and stain project. There is something about a grill in burnt wood that just looks good. There are a lot of ways to build them. I would recommend using a miter saw for most of your cuts if you can get one or have one. Just makes it super quick to measure and cut at a 90 or 45 degree angle.
Honestly, I would just go with some pine wood. Cedar will cost a lot more and if you plan on staining it I don't think it will really make a big difference appearance wise. The stain should last a pretty good amount of time if it isn't getting hit with rain/sun all the time, but be careful how close you put it to your house as it could stain/melt siding.