Leave your firring strips in place, but make sure they are fastened tightly to the joists at the intersecting points. Plaster as a rule is 3/4" thick including the lath, although it can vary, especially in turn of the century homes. The firring should be 3/4" thick too. You will want a minimum of 1" penetration into the framing to secure the firring to the joists, so 2 1/2" to 3" screws should be ideal. You will not want to go over kill on the length for you may hit something in the ceiling you may not want to. As I mentioned before 1" into the framing will hold anything in the way of sheetrock you install. Pilot the screws through the firring only will help.
After that you can install the sheetrock any way you choose. I have run it with the framing, I have run it across the framing. Either way will hold fine, though some people may beg to differ. But hanging drywall is not rocket science. Set your screws the proper depth, which is critical. There are special drive tips that govern the depth for you.
As note, now is a good time to update the wiring in the ceiling PROPERLY, but you may need to open walls to do so.
If the ceiling height is tight, but I think you may have high ceilings anyway, you can remove the firing strips, which will let a lot of plaster fall, (not good) and screw the drywall straight to the ceiling. You will need to find the joists to some degree, although the lath is capable of supporting SOME of the weight, I would not rely on it solely. This would take approx. 2" screws to hold this up. If the plaster is for the most part intact, except for a number of small holes from the nails, you can use 1/4" drywall to overlay the plaster.
For my opinion, I would stick with going directly over the re-secured firring strips with 1/2" sheetrock.