When installing a stone fireplace, do I install the wood mantle first?
To mount this mantle you will need to drill into the stone face!

The outline of the mantles' position should be taped to keep track of its placement. You will need to scribe the profile of the stone to the wood mantle. More on this if needed.

As long as the scribe marks are back cut severely, in essence, hollowing out (making it concave) the back edge where the mantle meets the stone, it should fit pretty well, save a little extra taken out here and there. You could use a small diameter disk sander on a drill with a VERY coarse grit paper on it. Self stick or screw on should work.

After you get it scribed where you want it to fit, find 2 or 3 suitable places to drill and epoxy steel pins into the stone face. Choose smaller stone to drill near, they will be better encapsulated in the mortar than larger stone so they have less of a tendency to fall out when drilled. 1/2" holes, 2 to 3" deep with 3/8" pins straight in with epoxy should do there. Please do dust out the drilled holes, the dust will not let the epoxy bond. Let it cure. Concrete epoxies only take about 15-45 minutes to get hard.

On the wood mantle, drill 3/4" holes, you can fill with the same epoxy, even if it is for concrete, the wood won't mind. Concrete epoxies offer two mixing tips, this will come in handy for the setting of the mantle.Here is an example