A power drill with a 1/8" bit stopped at 1" "should" help you locate your "studs" (in the ceiling, you would call them joists if they support a floor, or rafters if they are part of the attic) keep in mind that they ought to be spaced at regular intervals, (usually 16"oc for joists, vs 24" for rafters) If you chalkline your framing members on the ceiling and continue the mark a little onto the walls, you should be able to see what you are doing really easy. Lath is NOT strong enough to support your drywall.
3/8 drywall should be fine, assuming the ceiling you are attaching it to is on a flat plane. Any bows or wows are going to transfer right through, and you can adjust for that using shim strips (of cardboard) that you can obtain from your drywall supplier or most hardware stores. The strips are usually about 4' long and as wide as a stud.
You will need to use longer than normal drywall screws to attach the board, and the spacing should be no farther than 12"oc for the screws on each framing member. DO NOT USE NAILS!!!
You can use a trick to hold the board up while you screw it in. Build a letter "T" out of 2x4's that is 1" shorter than the new ceiling height, and use it to support the end of the board away from where you start to screw the board in, then work your way down to the T. (alternate method, invest in a case of beer and pizza and invite a few friends over...)
One last note. You might consider getting a bid or two from the pros on this one. Ceilings are notoriously unforgiving of unprofessional work, and very few pieces of furniture are capable of hiding the errors. Light reflects across the surface of your work and shows EVERYTHING! This is not really a project for the faint of heart, and if you shop around, you might find it's worth it to A:) have the whole job done by a pro. B:) have a pro do your mudding taping and texturing C:) a variation on the above.
Be sure you check out your drywall pros. The actual level of professionalism can vary HUGELY, and is never reflected in their bid price or the type of van they drive. Check them out with their suppliers, and ask for at least 2 references.