Okay, I have to say one thing here, which is something that's been bothering me a LOT for the last few weeks. I am not pointing fingers at anyone specifically; this is a general thing I've noticed across the board on my part.
How we feel about the spay/neuter/adoption/breeding issues on animals (any breed) is not the question that was being asked here by the poster of this question. The poster was not asking - is this particular animal breeding quality, nor were they asking for our opinions on we feel about the morals of breeding purebred cats, or if purebred cats should be bred. They were asking if any of us had any bred cats and what tips we might have, for education.
That said, my family raised and showed cats (I started showing when I was four) and so do I to this day. I raise Korats, and yes, I've won Nationals with my cats. I was lucky in that one of the top breeders in the United States decided to mentor me when I was young, and I learned from her and her cattery policies, and the way she raised cats. I'm honored, frankly, that she put enough trust in me to think I could live up to her standards.
As always, I stick a disclaimer on my posts like this. A lot of this is only my opinion on things. It is an opinion, nothing more, nothing less. It's just what my suggestions would be when a person is considering raising purebred cats.
1) First off, you need to have a mentor, because otherwise, I'm sorry, you're NOT going to have good breeding stock to start out with. Nearly all of the top breeders do not sell our breeding stock unless it is to someone we know and have worked with, and know our cats will be treated/bred as we would.
When someone tells me, "I want to breed Korats, so I'd like a cat from you," if I don't know that person's been involved in the breed (showing regularly, etc.) I'm NOT going to sell them an open cat. Sorry. Call me a jerk, but I don't want my breed or my cattery name out there for a fly by night person to sell or breed and try and make a quick buck.
No way. I spay and neuter all of my pet cats before they leave home (and yes, that includes my own mother's cat.)
2) Get into showing for a few years before you even start. I cannot stress this enough. There is no reason to work with a breed of cat otherwise. Unlike dogs where working and show-line animals sometimes have very different standards, cats really, honestly, truly, are either show quality or not. We don't work cats, or train them to sniff out bombs. Their working abilities don't tip them over to breeding quality. It's all about - does this cat match up to the breed standard, and does this cat exceed its parents in those standards. If they don't, they need to be spayed, neutered, and placed in pet homes out of your breeding program, and you need to think about doing the same to your adults.
You need to know your breed standards, you need to know what a great example of that breed looks like, and you need to know who has good tempered, friendly cats and who doesn't. That only comes with getting out there and being in the show circuit for a while, I hate to admit it.
Unlike most dog shows, cat shows have "Championship" status for cats that are neutered and spayed. It counts towards your ROM with your queen or tom. My Nationals cat, Luke... (well, when he wasn't retired and fat, ha ha) is altered, even though he would have made a gorgeous tom, I was NOT at the time in a position to keep an unaltered stud. I wanted to show him and put points on his mom and dad, though, for both me and my mentor. Now, I personally prefer showing in the altered classes, myself, and I've often sent off my whole cats to my mentor to get them finished - my buisness grabs my weekends and chokes them.
Having someone else handle your cat for you isn't cheap, though. Having them do it well especially isn't cheap, but heck, it was worth it for me for one of my cats; shows are California one week, Idaho the next. I just don't have the time sometimes to do that.
3)KNOW the health problems in your breed. Research them. Research your pedigrees; just like dogs and horses, cats have lines that mix and don't mix. And certain lines have certain issues that others do not.
4) Genetic test and disease test your breeding stock. Period. Can't stress this one enough; there are lethals out there in the cat world, and serious genetic disorders, just like dogs and horses. Like joustingirl said, there is so much transmittable that can spread through a cattery like wildfire, and if you sell a cat with these diseases to someone else... that's just not fair to them or their cat. Period.
5) Vaccinations are not created equal. For example, my cats cannot use the modified-live versions of vaccinations. I have to special order in killed vaccine for my Korats for my area. I lost a kitten to distemper because she actually picked it up from the vaccination. But you have to vaccinate and keep accurate records.
6) Make sure every square inch of your cattery area can be disinfected. Joustingirl is right; a lot of the cat diseases are airborne and contact, and some of them are downright deadly. You've never lived until you've run bleach through a carpet cleaner and your house is several shades lighter.
7) Whole cats usually mark. Actually, I had more problems with my girls peeing all over everything than my boys. *laughs* And cat pee stinks, it ruins your carpets, it ruins your house's resale value, and boy oh boy, can it ruin your relationships... that was okay. though.

I didn't like the guy that much, and the cats were far more faithful. *grin*
8) Socialize your cats. I handle my kittens, they're underfoot in my house with the dogs and friends and other cats. I handle them, I flip them over, I kiss their tummies, I touch their paws and clip nails, clean ears, clean teeth, everything. I treat my cats like I treat my dogs (yup, even take them for car rides in the car in carriers) , and my cats are easy to handle and friendly. I personally don't like kittens raised in cages; it can be done, but I like my cats to be house-savvy. I have a baby-proof kitten room that they go back into when we can't watch them, but most of the time, they're out and being obnoxious. XD
9) Spay and neuter before your kittens go to new homes. Now, I know that this is difficult in some circumstances for some cat folks. My kittens do NOT leave my house until they are six months old. They are spayed, neutered, and microchipped here, with my information on the cat. I always keep my information on the chip as well as the new owner's information. I know that I can always be found, and if one of my cats was to ever end up in a shelter or bad situation, I would darn well fly across the country to get them back. I bred that cat, that cat is my responsibility for life.
10) Contracts. I can not stress this enough. Have a sales contract and have it drawn up/looked over by a lawyer. No loopholes.
A few points that I have in mine are - I do not allow my cats to be declawed. I do not allow my cats to go to homes which allow outdoor access without restraint. All of my cats must be vaccinated or have titer checks for anitbody levels. If the owner no longer wants the cat, the cat is to come back to me first. They can NOT give the cat away without permission to anyone, they have to contact me and I have first right of refusal. If the owner dies, the cat is to come back to me. Sounds harsh. Sounds like I'm a snotty cat breeder. No, sorry-- I just want them to be safe and even though I screen every family and I'm really picky... accidents happen. Even to the best of us.

Since all of my pet cats are spayed and neutered, I don't have to put that requirement in my contract - friends of mine do.
I also am a co-owner on all of the cats (well, one cat) that belongs to another breeder, and she is a co-owner on the cat she sold me. Both of us have to sign off on any litter registrations that occur from these cats. I think co-ownership often is a good way to get a higher quality cat from someone who normally would not sell you their cats... not always, but it can help.
Unfortunately, as far as advertising goes, I can't help you out. My mentor has a four year wating list for kittens, and she has graciously shared her people with me when she felt my kittens were better suited to the households that were applying for her kittens. I have found, however, throughout the years that well established breeders with good stock are usually not the ones who advertise in the classified ads.
Last but not least, and this is where I try to walk a line between opinion and personal point. If you're breeding cats to get a return on an investment... all I can say is.. I have yet to break even. I never breed a litter to think I'm going to make money - I breed a litter to better my breed. It's a hobby, not my business, and I don't think I've met very many excellent breeders who didn't have a second income from another source.
In fact, I think I'm about fifteen grand in the hole, thank you very much. *laughing* I finally threw away my Quick Books program after my significant other said, "Oh MY @#$# did we spend 7,000.00 on the CATS THIS YEAR?"
It's better if we just don't know where the money went... Well, better for me. The cats don't mind if I'm sleeping on the sofa. It's like a nice warm place for a pile-up.
I hope this long answer was useful... You were sort of general and broad in the question, and I'm hoping I helped with some food for thought and suggestions on what to do, and what not to do.
*Edited simply because Dane pointed out to me I shouldn't number my points when I'm tired. *laughs* Ah well.