Restoring the bike is a great way understand the bike.
My history is similar to yours. Gave up bikes around 10. Wife want to me ride with her.
I got a top of the line bike on CG for peanuts. Of course, it was top of the line around the 90s :)
Start by REALLY cleaning the bike. Simple green and brushes. Don't be afraid to wet the bike. Just avoid the seat if it's leather. And no water inside the tubes.
I'd start replacing the shifters for indexed ones. Getting the same brands will usually work. But will require a lot of research to match them with the rear derailleurs (the gear shifting thing on the back).
Another option is to just keep the ones you have (friction shifters) and feel the gears changing. Even with modern bikes, this is still the norm with the front gears, so no harm on going that route also. Plus, restoring and fine tuning the bike your self will make you 'know' how to feel the gears better.
UPDATE: I take that back. I kept my friction ones on my bike and rode the wife's bike with cheap indexed ones a few times. the friction are way better. also when the freewheel needed replacement I can get any number of gears I feel like and don't worry :). Indexed ones are better for changing on a whim. but are either freaking expensive or will require constant fiddling. friction are just install and forget. love them. Unless you go expensive indexed. But I'm a cheap bastard.
If it has sealed bearings in the middle of the wheel you are fine. Those last a life time. It's that axle on each wheel. Just turn them. If they keep rotating without rattling noises, you are fine. If they just turn a couple times after you roll them with your hand. You need new ones.
Also, if they wobble when you turn them, they need truing. That's the only skill that does not pay off to learn yourself. $15 at a bike shop and the wobble is gone.
If they rattle, don't spin, wobble, just get new wheels.
Inspect them all. Pull them with your hand via each housing. If there's any drag, try to fix it (I did fix my rear derailleur cable by cutting one broken thread that was pointing outwards and so dragging in the housing. good as new now). If it's a brake cable, spew out the $7 and get a new one with new housings.
WD-40 into all cable paths also help.
Begin by checking that they still have plenty of rubber.
Also check your wheel rims for damages.
You have to loose them and tighten them as close as possible to the rim. in a way that they are slightly curved against the wheel, with the BACK of the brake touching the wheel. the front of the brake is the front of the bike. It should look like: \ || /
You should be able to touch the rim with some <15% of travel of the brake. The rest of the travel should be extra pressure for breaking power. Play around a lot with this. There's several places where you can adjust your brake.
Second step is to adjust them in a way that when you press the brakes, both sides touch the wheel at the same time! There's usually a screw on the side to control that.
Make sure that NO other cable housing goes UNDER the front brakes cable.
And I can't stress this enough: test drive the bike on a straight run EVERY time you adjust them. Mostly now that you are getting to know things.
A bike has serviceable bearings in the cranks and in the freewheel (the thing that rotates the back wheel when you are not pedaling). Usually your freewheel will be dry, cranks are almost always fine. check it by rotating the wheel and leaving the cranks still. If the chain is dragged to the front at slow rotating speeds, you have to grease/change the freewheel. I'd only change if the cogs are worn out.
You will need a spanner tool to open those.
You will hear about them a lot. It's the pair of gears on the back that change the drive gear on the rear wheel, and the guide on the front that change the drive gear on the cranks.
It is the easiest part to restore and adjust. If they do not move sideways when you press with your hands, open them up, clean, and grease the hell out of them. It's easy and was what took less time for me, but everyone will say the opposite.
When they are moving easily when you press them, read this parktool's article on how to adjust it.
There's probably more than I can remember now... but all in all it took me around a week and a half. And around $50 in tools/greases/new chains. I didn't have to replace any component or cable yet. And now I love to ride. I go to work every day riding.
Now, to fully understand what you are doing, read:
The last link will point you to several expensive tools. I only need them to work on the cranks and remove the freewheel (you don't have to remove the freewheel unless you will replace it, I didn't know it back then). I got the tools asking around at work, one guy had a nice kit I could borrow.