It's not very cheap and is likely comparable to dog grooming costs.
However, the issue with cat grooming is that there aren't many places to compare prices in my area since most places don't handle cats. This leaves only veterinary clinics, and even then, most will do the fur and nails but won't wash the cat.
Grooming also involves the additional cost of sedative medication, which cost more than the actual fur cut
The last time I had my cat groomed, I paid a little under $200 AUD, but prices can vary, but you'll need to check your area to see what options are available.
Typically, at the veterinary hospital where I take my cat for her health needs, they only do her fur and don't provide washing services.
For bathing, I have to take her to another veterinary clinic in a different area.