Since the early 70s bike boom, components and standards have evolved considerably, but old steel frame bikes still live on. In fact, many are better built and nicer to ride than most new bikes. These are the guidelines I follow.
1. Brakes
Brakes are the most important part of a bike. If you can't stop, well, that's a big problem.
Almost all vintage bikes use rim brakes. Older bikes usually use centerpull brakes, which have a wire yoke connecting the two brake arms across the top. These are finicky to adjust but work great. Newer bikes use dual pivot brakes, which are super easy to adjust and also work great.
Bikes made in the late 70s and throughout the 80s, however, often used sidepull brakes, which are not my favorite. They tend to be weaker than centerpulls and dual pivots and are prone to misalignment, where one brake pad insists on rubbing against the wheel rim. I never use sidepull brakes, even nice ones, and will usually upgrade to modern dual pivot brakes on a restoration.
As for brake levers, most are super simple and work just fine. For some reason, however, mid and low end road bikes always came with brake lever extensions that allowed the rider to brake while riding with the hands on top of the bars instead of on the brake hoods or drops where they belong. Also called turkey wings or suicide levers, these extensions are trash. They diminish brake pull strength and are prone to failure. I recycle them all.
I also make sure brake pads are still in new condition, which means I usually replace them with new. Brake cable housing can often be reused, but I always install new brake cables as well.
2. Kickstands
Kickstands are dumb. They rattle when you ride. They add weight to your bike. They often fail and your bike tips over anyway. We should all just lean our bikes against a tree or lay them down. I recycle old kickstands.
3. Wheels
Lower end vintage bikes often came with steel rim wheels. Steel wheels are heavy and they don't brake well when wet. If an old bike is worth saving, it deserves alloy wheels. I recycle all steel wheels.
Old road bikes also came with either 27" wheels or 700c wheels, two slightly different standards. Today, 700c wheels are far more common and it's possible to convert a 27" frame to use 700c wheels. However, there's nothing wrong with an alloy 27" wheelset and it's still possible to buy nice new 27" tires from Continental, Schwalbe, and Panaracer.
4. Bearings
All the moving parts of a bicycle require grease and bearings to move freely. Over the years, the grease gets crusty and bearings wear out. Every vintage bike I service gets a fully rebuilt headset and bottom bracket with new grease and bearings. Often, the wheel hubs require new bearings as well. I use loose bearings instead of caged bearings because they spin better.
5. Grease
Grease isn't just needed for bearings, it's also necessary to prevent parts of the bike from seizing together, especially if the bike is ridden in rain or snow. The seatpost, stem, and every bolt should all be treated with a bit of grease or terrible things will happen.
6. Chains
A restored bike needs a new chain. A good quality KMC chain isn't expensive. Every bike gets a new chain.
7. Shifting
Derailleurs can be a bit tricky, but there are a wide variety of vintage and new derailleurs that work great. The list of crappy derailleurs is too long to list, but I have have a particular disdain for Simplex derailleurs, which were often used on otherwise very nice French bikes. Simplex mechs look kinda cool, but even the best don't perform well. Simplex also used plastic delrin parts, almost all of which have cracked. A vintage Shimano or Suntour derailleur is vastly superior.
Likewise, many newer bikes use plastic twist grip shifters, trigger shifters, or "brifters", where the shifter is integrated with the brake levers. All of these are prone to failure. I much prefer bar end shifters (barcons), thumb shifters (thumbies), downtube shifters, or even stem shifters, all of which will perform perfectly for ever and ever.
8. Cranks
Life's too short to deal with cottered cranks or Ashtabula cranks. Both are usually, but not always, a sure sign of a low end and heavy old bike. Stronglight invented the square tapered crank in the 1930s and to this day it is the best method of attaching crank arms to the bottom bracket.
9. Gas Pipe Bikes
Speaking of low end bikes, the vast majority of bikes sold in the past 60 years really aren't worth saving. Models like the Schwinn Varsity, Suburban, and Continental vastly outsold lightweight models. Other common examples include the Peugeot UO8, the Raleigh Record, and almost any bike by Huffy, AMC, Murry, Sears, Montgomery Wards, etc. Today, although some people love and even collect these old heavyweights, they're really not worth restoring as a riding bike.
New department store bikes are just as bad, even when they're made out of lighter weight materials like aluminum. The components on department store bikes, especially the brakes, are so outright terrible they ought to be illegal.
10. Stems
Finally, a word on vintage bicycle stems, which attach handlebars to the steerer tube and fork. Stems are an important part and they carry a lot of weight and torque.
Vintage stems were made in two styles, one of which used a metal wedge (usually steel) to secure against the inside of the steerer tube. The other style used a cone that expanded inside the bottom of the stem to secure against the steerer. Cone stems were more common on European bikes than Japanese or American. Because the cone shape requires bending the alloy metal, they tend to develop stress cracks. Some vintage French cone stems were particularly prone to failure. When I'm restoring a bike with a cone stem, I always replace it with a similar wedge stem.
I probably have more than ten opinions about old bikes, but these seem to be the most important. Let me know in the comments if you disagree with anything or if I missed something important. Thanks!