I'll share my personal formula, which I use myself and in a couple bikes I often perform maintenance to, and that are ridden by other people. These tips are for V-brake or "linear pull" type, but the same can be applied to other types too, at least caliper and cantilever, as long as they are cable actuated (i.e. no hydraulic).
First of all, I check brake pad alignment. This step is critical for good braking performance, and this bit is the one that can affect your noise problem. It also affects the perceived "correct" distance between pads and rim, but this topic slightly falls outside your question, so, I'll just assume you have your pads aligned.
In most a linear pull (v-brake), cantilever or caliper type brakes, the distance between pads and rim is determined by the "length" of brake cable relative to the length of the casing, adjuster barrels, cable stops and other items in cable routing. Cable slack increases brake lever "travel" which means how far you have to move the lever until it effectively starts braking. This distance is the critical parameter for me, this is personal because it depends on your hand size and the combination of lever/grips you use. Also, your hands have a natural point in which they "squeeze harder". This is the point where it takes less effort for you to actuate the levers. Your levers should move so they begin the real braking action just a little before you reach this "hard squeeze" position. This way you have your hand in optimal position when you need to brake hard.
I start by screwing down any adjusting barrels or bolts to a minimum (or near minimum). This will loosen the cable a bit, increasing pad to rim distance and also the "travel" of the brake lever, this means you'll have to press deeper the lever for the brakes to actually touch the rim. These barrels are usually located between the lever and the cable, or between the cable and the caliper (if any).
Now I loosen the bolt that holds the cable to the caliper or the V-brake arms, and while pressing the pads against the rim, pull the cable so there is no slack, and tighten the bolt.. but just a little! only enough to avoid the cable from slipping out from just the spring that separates the pads from the rim. Here is the trick: with one hand, hold the cable between your index and thumb, in such manner as if you loosen the bolt again, your fingers will prevent it from slipping out. Now, slowly depress the corresponding brake lever until you almost reach your "hard squeeze" distance, meanwhile, the cable will slip between your fingers, but the pads should remain pressed against the rim. Without releasing the cable, tighten the retaining bolt again, this time tighten it fairly tight. Check the lever travel and give the final tightening to the retaining bolt. Repeat for other other lever if needed.
Now spin your tires in the air to check whether it rubs the rim. Re-check any bolt you have loosened for proper tightening.
Don't panic if you don't achieve the perfect adjustment in the first try, you have a second resource: the adjusting barrels.
The adjusting barrels have two purposes: One is to fine tune this distance. They work by taking up or giving out cable slack. Usually they consist of a "hollow bolt" the cable goes trough, and is bolted to the lever assembly or the caliper. The other end of the bolt receives the casing. By "unscrewing" the barrel, they add distance between the casing and the lever or caliper, thus reducing cable slack. Almost all these adjusters have also a locking nut that you screw all the way down to the lever or caliper, so it prevents the adjuster from moving "on its own".
The other use of this is to re adjust the brakes as pad wear increases the distance between rim and pad surface, so you keep the brake lever travel in its optimum.
In an ideal setup, this is it. But there are other considerations:
Rim truing and dishing: The rim should be straight and properly centered. Untrued rims will rub the pads in certain part of a wheel revolution, but not all of them. A rim that's not correctly "dished" (i.e. centered) will rub against one pad all the time. V brakes can be easily adjusted toward rim centering because they have two independent return springs, each with an adjuster bolt that regulates it's tension. By balancing both sides' tension you get the pads the same distance from rim. Also a deformed rim may have bulges or cavities that complicate things further.
Cable condition: cable and its casing should be in good condition too, not rusted, not overly worn, not sharply bent.
Bike fit: Levers should be positioned properly, according to bike style, riding purpose/technique, sizing and personal preference.
Component combination: In some cases, certain combination of specific components, brands, designs, can make it very difficult or even impossible to achieve correct adjustment. This is very unlikely to be your case, as you stated your brakes worked fine before.
As you see this is a practical approach that doesn't deal with millimeters or such. When you finish go for a safe test ride and repeat the procedure if necessary. The first times you'll net a few tries before you conquer it, when you get used to it, you'll be able to do it in one attempt and without needing to move the adjusters.