Make sure to have a plan in effect in the case of a detector going off.

One on the bedroom level, in the hallway outside the rooms, one on the main level, not in the kitchen, and one in the basement, near, but not next to, the furnace and water heater.

In our house, there is one more, at the top of the basement stairs above the door to the kitchen. That one occasionally goes off when we're cooking. I open the kitchen slider and then fan the basement door like a bellows. That moves enough air to clear the detector and stop the alarm.

I'm going to install CO detectors in all three houses too. No requirement, but it is the right thing to do.

There are basically to types of detector technology, ionization and photoelectric. I switched the one near the kitchen and greatly reduced the number of nuisance alarms when using the oven and the stovetop. YMMV.

and I used to wrap it in a towel when I baked anything because just preheating the oven would set that sucker off.

They're really best in hallways outside bedrooms, IMO. You're generally awake in other parts of the house.

The only thing I fear is wiring. It's marginal in this dump. The wood stove is usually just about out when I go to bed, I don't smoke and I never leave a stove on overnight.

quite old.

my daughter in law's sister had a bad fire because they didn't clear the dryer lint screen! I always clear it after each dryer load...

it had been 15 years and I had never cleaned out the vent hose. So I crawled in back of the dryer and disconnected it and found there were only a few stray wisps of crud in there, I was seriously impressed considering the cat hair in this house.

Cleaning the lint filter every time or even every other time is key, I guess. That hose was nearly pristine because of it. I did run a perfunctory swipe of a duster through it, but got very little out of it.

Twice I've needed one. It was no fun being without.

Once when I was still living with my parents. My mom was braising some beef on the stove and the oil went up like a roman candle. Getting a lung full of smoke from this relatively small fire was no fun either - now I see how people can keel over immediately from smoke.

Another time I stopped to help some kid tossing snow on his engine fire. All I could do is call the FD and help with the snow tossing. By the time the FD arrived, the car was fully engulfed.

I keep a small extinguisher in the kitchen and a large one in the living room for the fireplace. I keep another one in my car.

Also, I installed these in both the condo's store rooms and smaller ones near the electrical boxes and one behind my gas dryer:

From Griot's Garage magazine:

http://www.griotsgarage.com/product/ceiling+mounted+unmanned+fire+extinguisher.do

but let them just burn down) and then went to a friend's house for dessert one Friday night. Her house burned to the ground! You are supposed to put the shabbas candles in the sink to let them burn down.

Personally, I would not light a single candle in my house. But I did leave a burner on low on the stove and it singed a pot holder I had lifted the pot handle with. The house filled with smoke but I was on the porch and realized what was going on. That scared me.

At least 1 on each floor, including basement and attic.
1 inside each bedroom - and the bedrooms ones do not count towards the "1 on each floor" rule.
1 in the room at the top of any stairs
In multifamily houses, you also have to put one on each level of any stairs.

And in a new house, they'd be wired together so that if one goes off, it sets off the others.

As for fire dangers, generally speaking the stuff permanently attached to the house is built with some safety features to prevent a fire. Of those devices, cooking devices are the most common cause, followed by dryers. Fireplaces are usually not a cause. They are generally rarely used, and even when they are, they're designed with fire in mind.

Most house fires come from non-permanently-installed stuff, like cigarettes, candles, space heaters, hot plates, and so on.

Smoke detectors are generally placed away from the kitchen unless there's no better place for them. If they're near the kitchen you'll want one of the "photo" smoke detectors, because they are less sensitive and thus less likely to false alarm when you burn dinner.

My plan is to put one in the basement (between the utility room and the finished basement room) and one at the top of the landing of the second floor, between the bedrooms. There is very little space between the two bedrooms and the little office.

On the first floor there is a living room, dining room, kitchen, powder room and a den (which was an addition). Maybe place one in the living room?

Put the basement one at the top of the stairs: Smoke rises. So if there's a fire in your basement, the smoke will be at the top of the stairs. It will take a while for the smoke to build up enough to trigger a smoke detector on the basement ceiling. And you don't want that. You can put it on the wall of the stairs, as long as it's within a foot of the ceiling. That will make it much easier to change the battery than if it's on the ceiling.

Outside the bedrooms is a good choice for a retrofit. Many states require landlords to put one there on rental properties, if there aren't smoke detectors in the bedrooms. In new construction, the "inside bedroom" detectors serve two purposes:
1) if you're asleep and the fire starts in the bedroom, it's good to be woken before smoke builds enough to trigger a detector outside the bedroom.
2) Since the detectors are wired together in a new house, the in-bedroom detector will wake you if a detector far from the bedroom goes off.

As for the first floor, I can't give you a good answer without seeing your floorplan. The rule of thumb is to locate the detector as centrally as possible. This runs up against the "away from the kitchen" rule.

Since any smoke will rise, placing a detector near the bottom of the stairs won't help that much, because any smoke would set off the 2nd floor detector anyway.

Finally, if the kitchen is a separate room (the only connections are through doors or doorways) then the wall over the door will take care of false alarms, and you can put the detector anywhere outside the kitchen. If the kitchen is open to other parts of the house, look at the ceiling. If there's a beam between the kitchen and the other rooms, that beam can also catch smoke and reduce false alarms.

One thing you could do is just try it near the kitchen. If you get too many false alarms, you can always move the detector.

But the stove is right up against that wall. I am thinking perhaps putting it a bit further down the basement steps so as not to be so exposed to any smoke from cooking.

Assuming the door to the basement is normally closed, smoke isn't going to trigger a detector on the basement stairs - the smoke will rise to the kitchen ceiling. To get to the basement, the smoke would have to be thick enough to fill the ceiling down to the door, and then have a high enough concentration to work through the very small gaps around the door.

What would? Would it be a dining room (next to the kitchen, natch) or a living room smoke detector? That is, assuming that the next smoke detector would be in the small hallway between the upstairs bedrooms?

You're looking to balance going off with actual fires and not going off with "whoops, I burned dinner".

An actual fire will put off plenty of smoke, and will trigger all the detectors in the house...eventually. The goal is to slow detection from the kitchen to the point where you can open some windows if it's the "burned dinner" case.

I haven't done it yet on my new kitchen stove because I am afraid of it. I now know it is not going to burn a hole in the kitchen floor (my big fear!) so I will do it. But when I do (and there's a lot of crud in that oven) will the burning up of it cause lots of smoke?

Just as important, especially in Winter when your furnace is going.
Unless the smoke d's and co2 d's start to beep beforehand, I change the batteries every Christmas.