Oh boy - the Southern Tier of NY from California - boy are you in for a change! I grew up there, moved away for a decade or so and moved back in 2000, just a bit further north. That area is a nice place to live but a bit off the beaten path. Lots of great hunting if you’re into deer and ducks. You’re in luck if you enjoy fishing too. Rochester is a short hump up 390 but there is a "mall" and a large shopping complex over in Big Flats which is +/- 40 minutes east
Slowly but surely, NY is making the process to get your permit faster. As peetzakilla says, you need to be a resident of the state for 6 months and come up with four references who live in the same county. The fees add up to about $120. I'm not sure about Steuben county, but many counties have gone to a computerized system where they take your finger prints on a scanner and send them off digitally so your background check is very quick. Your longest wait will be your paperwork sitting on a judge's desk. It took me just over a month from my Sheriff's interview until I had the permit in hand, not bad considering it used to take somewhere in the neighborhood of 6 months.
To carry concealed in public though, I think the process might be different. Maybe some other FL bubbas can chime in here to set me straight, but I think to actually carry on your person in public, you have to come up with a good reason you need to do that. My permit has restrictions where I have to lock the gun in my trunk or have it locked in a case to transport it to the woods or the range, but it's still considered a CCW..strange isn't it. From what the Sheriff told me when I interviewed, the process to actually carry concealed requires more paperwork and takes longer. I'd give Sherriff Tweddell's office a buzz and see if they can give you the straight scoop, If their website it up to date, his number is (800) 724-7777.
Good luck to you sir!