Disclaimer: I've done this exactly once.
Here are some things I did in the last few days of renting out my single-family house.
Go around to all the rooms and make sure everything works - doors open and shut, windows open, shut, and lock, mini-blinds (if any) are securely mounted and work, curtain rods (if any) are securely mounted, closet rods are securely installed, none of the light bulbs are burned out, etc. If you were thinking of oiling a door hinge or putting a new screw in the curtain rod bracket - now's the time.
If you have mini-blinds that have pull strings to operate them, take the time to trim the strings such that they don't dangle to the floor, and install a cleat on the wall next to the window so the strings can be wound around the cleat when the blinds are pulled up. This keeps little kids from getting tangled up in the cords.
Make sure all the appliances work - mostly in the kitchen, but also things like a garage door opener or sump pump, if the house has one. Run all the sinks, flush all the toilets, run the shower and tub to make sure they are OK.
Put fresh batteries in anything that needs them - smoke detectors, doorbell, garage door opener remotes, closet lights, etc.
Make sure you have enough door keys for both you and the tenants. If you haven't changed the locks since you bought the property, do so! If you have to get more keys cut now, take all the keys to the house and try each one in all the locks, several times. This prevents "I went to use the second key and it doesn't work" problems. If any of the locks are stiff, lubricate them - my locksmith recommended "Tri Flow" spray, which I used to good effect.
If the house has a garage door opener, gather all the remotes you have, tell the opener head to forget all the remotes, and then learn in the ones you have. (This prevents former occupants from accessing the garage.)
Do the last cleaning on the house. You probably want to do this as close as you can to when they move in. Bring a stepladder and a flashlight and clean all the nooks and crannies that nobody ever cleans.
Do the last yard work on the house, if it has a yard - mowing, trimming, anything like that. If it's a house in the city with no yard, maybe sweep or wash the stoop and sidewalk. Pick up any trash in the yard, along the street, etc. This should happen as close as you can get it to when they move in, but it's OK if it happens 3 or 4 days before.
If you don't already have them, write down the model and serial numbers of all the appliances. Take a picture of the model number label, if you can. This helps in case you need parts later, or in case an appliance is removed or swapped out.
Get whatever things you want to have in the house at move-in. At my rental, I left a new roll of paper towels, a bottle of Windex, a new bottle of hand dish soap, a new bottle of hand soap, a 4-pack of toilet paper, and a pack of clothes hangers in the house for move-in day. The bathroom stuff ensures that they can use the bathroom, even before they move anything else in. The kitchen stuff helps them clean up after move-in day fast food. :)
Get ready to switch over any utilities. With water, gas, and electric, usually those will stop on their own when the tenants get them put in their name - but be sure to watch the bills to make sure they do this. I had to call the trash company and stop the service myself, because the tenants selected a different one.
Does the city where the property is require you to have a business license, or to register your rental property? You should have already found out about this, but if not, try to get legal this week.
To address a couple of specific things:
> radon testing
Either this is OK or it isn't, and if it isn't, you will be asked to buy an expensive blower system. For some reason, the companies that sell tests *and* blower systems always think you need a blower system. The test can be done in the next week, but scheduling the blower install may not be possible in that time.
> number of smoke alarms per unit
This is a building code thing. The minimum standard is one per floor. The next step up from that is one in each bedroom, plus one in the hall outside the bedrooms. The next step up from that is one in each place where people would conceivably sleep (like in the living room, for people that sleep on the couch). Some cities will require that new ones be the type that are wired in to the 120 V AC and have a 9 V DC battery in each detector. These are usually also wired together, so if any detector smells smoke, the horns on all the detectors sound.
If you don't have enough detectors, and If local building code allows it, you can easily get battery smoke detectors at the hardware store or even Wally World and install them in a few hours. If the wired kind are required, and you don't know how to do it yourself, you'll need an electrician, which may be a challenge to get done within a week.
I hope this helps!