The answer to this depends on 2 questions. How historically accurate do you want your house to be? And how much do you wnat to spend?
With lath and plaster, first you put up the wood strips, then a coat of plaster is added and forced into the space between the strips so it acts like keys to hold the plaster up. Finally, 2 more coats of plaster are added, the final coat of which can be smooth or textured. Repairing this is a job for a pro, not because of cost of equipment but to insure that the job is done right, which takes experience. Ceilings are worse that walls and if not done right, you can be doing it again soon. If you hire it done, be sure to ask a lot of questions about how many jobs the pro has done and how they have held up. Don't hire anybody young and cheap for this.
Now the other way to go. If you are not concerned about historical accuracy and cost is your main criteria, put up drywall. This will be at least a two person job. If you have to DIY, the best way to start is to keep watching home shows on TV until you find one where it is being done as it is a heck of a lot easier if you see it done rather than read about it. Briefly, use a stud finder and lay out the joist pattern. Make 2 dutchmen and put the drywall sheets up. Nail or screw them thru the plaster ceiling into the joists. Mud and tape the joints. This is a nasty, messy job and the same warnings go if you hire it done but there are a lot more people out there who know how to do drywall than know how to do plaster.