Daily: check the tyres and chain. If you need air or lube, apply them. This really falls under the "when riding, be aware of your bike". Rather than thumb-test my tyres I look at the bulge in the sidewall when I'm riding.
Weekly: pump the tyres, lube the chain. Regardless of whether they "need it", I'll put the pump on and check the pressure, then use a bit of lube on the chain. Swap rechargeable batteries in my bike lights (LED lights use so little current that it's easy to overdischarge batteries, and weekly is about as long as I can space this before I forget. Otherwise I think "did I do this last week? Probably"). Also pick the broken glass out of your tyres. An old spoke, sharpened at one end and bent into a handle at the other works well for this.
Monthly: fix the niggling things that aren't really faults that stop the bike working, so I never actually fix them. Lube and adjust the gear cables, adjust the brakes, tighten the mudguard bolts, repair or replace the loose reflective tape. Check brake pads for wear etc.
six months/seasonally: make sure the lights are all there and working. Check and lube the shifter on my Rohloff. Check girlfriend's bike and fix anything she hasn't bothered to. On derailleur bike (ick!) check chain wear. Check for frayed gear/brake cables.
annually: change Rohloff oil.
There's a whole heap of other stuff that I do when necessary, like putting new tyres on. If I get a puncture on a Marathon Plus I generally replace it, because that only seems to happen when they're quite worn. Lesser tyres I check whan I'm fixing the puncture (since I have to inspect the whole tyre to find the puncturing thing anyway). If shifting gets hard or braking reels funny I check the cables, since a fraying cable is often the cause of those problems. And so on.
Edit: what would a bike shop do in their checkup?
The main thing is that they are not you so there's a second set of eyes, and partly they have no interest in "oh, that's not right but it's a pain to fix so I'll just live with it".
So the mechanic will start with the obvious stuff I listed in my "evaluate a second hand bike" answer. The mechanic will be test riding the bike, so they want to be sure it's safe to do that. And anything that needs fixing, they will note down and either fix or ring you and ask if you want to spend the money.
It's probably easier to present as a checklist of functions. For each item mentioned there's an implicit "is it there, attached securely, in working order" check as well. This is all done in a proper work stand, with good lighting. Most of the tools are the ones you'll find on a compact multitool (but they're good quality separates). All cable replacements mean replacing the outers as well, and fitting end caps etc.
frame and fork. Check the bearings in the headset for slop and the grinding feeling of dirty or overtight bearings (crash damage sometimes comes through there when the bearing races have been pushed out of alignment or the frame is ovalised where it's supposed to hold the bearing race). Look for dents, holes or other frame damage. If in doubt pull out the frame alignment checking tools and go wild.
wheels. Spin them checking for bearing play and broken/loose spokes. Look for rim wobble too. True wheels if necessary in a proper wheel truing stand (they have to be true so we can set rim brakes later, and they're expensive to replace if they're wrecked which means sometimes the whole job stops here). Check rim and tyre wear and tyre inflation. If the valve is at a funny angle let the air out, straighten it and reinflate. Inflate tyres to marked pressure so later we can check for slow leaks.
saddle. Tight and in usable condition.
handlebars. Lots of stuff happens here, but for now just twist against the wheel to make sure the headset and handlebar clamp bolts are tight.
crankset/bb. Spin the pedals by hand and feel for slop/grinding. Check pedal bearings too. Eyeball frame clearance (is the BB the right length, are the cranks bent?)
gears. Change through all gears. Check shifters and cables.
- Replace cables and outers if there's any suggestion that this might become necessary in the next year. Poor cables means poor shifting and customers hate that.
- Pop cables out of the frame, slide the segments of outer along so you inspect the cable and at the same time lube the cables with a grease+solvent chain lube.
- Lube the chain. If the chain is filthy or worn (use a chain wear gauge), replace it.
- New chain means new cassette/freewheel. Eyeball the cassette anyway, if worn replace.
(Bike shops very rarely clean chains, it costs more than fitting a new one because of the time taken. If it's your bike you aren't paying for the labour so you would clean the chain at this point.)
brakes. All that gear testing means using the brakes. But now: squeeze each brake lever to the handlebars. This should take considerable force. Adjust the brakes so it does, the recheck. New pads and cables if there's any doubt (you'd be surprised how many cables snap at step one). Fine adjust brakes to just barely clear the braking surface. New disk rotor at this point if it's too bent.
mudguards/fenders, racks, lights, soft toys and other accessories. Make sure snything else on the bike is at least properly attached.
check the tyre pressure. It should still be whatever you inflated them to at the start. If this is your bike the check is unneccessary because you already know there's a slow leak and you will fix it now, won't you?
put the bike in the "to be test ridden" area for someone else to test ride (I lied above). They will check that the gears and brake work, and that the bike rides ok with no odd noises or anything. Mudguards especially can be a pain here because flimsy ones rub on the wheels at the slightest excuse. But yes, this is where you see the mechanic sprint to 30km/hr then do an emergency stop.
There are specific checks for other stuff. The rubber pad on the steering arm of Long John bikes gets replaced, roller brakes get greased etc.
If you're doing this at home you will probably need a spoke key to do basic wheel straightening on the bike where a mechanic will use a spoke tensiometer and truing stand. If you're going to send the wheel to a mechanic don't play with the spokes first. That just causes problems because your efforts will make the spoke tension really uneven. Fixing that takes time. You pay for the time...
A new bottom braket requires a crank puller and the right tool to grip the BB. It's not worth buying these for most people. It the BB has slop just get the shop to replace it.
Aside from those two most of the special tools that bike shops have make the job faster, easier and more consistent. You're never going to get "consistent" with one or two bikes, and it's not worth paying for tools you use once a year.
This should take ~30 minutes to check and lube, plus whatever repair time is required. At $100/hour charge-out rare it's $50. If your bike is clean and in good order it'll be significantly faster (5 minutes if there's nothing to do), if you bring in a rusty old "vintage classic" bike from the tip it could take days.