I wouldn’t turn it under. You will need the seam allowance to install the zipper. Frankly, I think this is optional, as the seam allowance will be covered by the lining. However, if your fabric is especially ravely or delicate, it may be a good idea to do it. It will help keep things a little tidier and easier to work with. If you dont have a serger, you can use a zig zag that goes right over the edge. My machine has an overlock feature, which is similar to a serged finish but with only two threads. I find it works pretty well for this purpose.
Ps – “neaten the seams” is Australian and British for “serge”. You do not need a server to do a nice job, however. I typically do a French seam below the zipper, flattening it out a little at the base of the zipper, which gets covered by the zipper tape.