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.
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