Rinsing is, usually, only the first step in brush cleaning. Many paints and varnishes will state on the label that brushes should be washed after use and if this advice is followed properly you'll never have a brush dry stiff or hard.
You don't need to beat yourself up about your brush having dried hard, it has happened to almost everyone at least once.... and for some it's a regular occurrence. I've rarely handled brushes other people have cleaned (and this includes numerous commercial painters) and not found them at least slightly stiff! Many have been much worse than this. Brush cleaning is a time-consuming and frankly boring task that comes at the end of the process where folks are already tired and want to just be done with it so it's very common for people to do a half-assed job. I'm guilty of this myself and I bet nearly everyone has been too.
Even if you thought you were rinsing the brush well you probably weren't using enough solvent, so in effect dilute varnish was left distributed uniformly through the brush head. The solvent container should be quite large in relation to the size of the brush for effective rinsing, as you want the dilution rate to be high. In practice this means that for a 100mm (4") brush you should be rinsing in something the size of a small bucket. Personally I think that even a small finish brush shouldn't be rinsed in less than 500ml of solvent, and going up to a litre wouldn't hurt.
In my case, I use my paintbrushes for applying varnish, however care for both paint and varnish (if there's any difference) would probably be most useful.
No effective difference. Although with paint it's much easier to see how successfully we've cleaned a brush since any paint remaining will stain the drying paper/cloth at the penultimate step.
The proper traditional method
Rinsing is just one step of a multi-step process in the traditional method, but it's not actually step one.
Wipe the brush as clean of paint or finish before you rinse. It's counter-productive to contaminate your rinsing solvent with a large volume of finish when you can so easily remove a large amount of it by simply wiping the brush dry beforehand.
Rinse in solvent. This does not merely mean swishing the brush back and forth in the liquid, you should press the brush out on the bottom or sides of the container to fan the bristles out and help the solvent get between them. Rinse a bit, wipe the brush against the rim, rinse a bit more, wipe. Do this a few times.
Dry out the brush on paper or cloth (it won't end up dry, you're just soaking out excess).
Rinse again as above.
Dry the brush as thoroughly as possible on more clean, dry paper towels or cloths.
Now you wash the brush in warm soapy water, splaying the bristles out in the palm of your hand or against the base of the sink, working up a good lather, and then rinse thoroughly. I can't stress how much difference warm water makes here. You can use cold where and when necessary, but warm water makes the job much much easier.
For the soap a block of basic hand soap is fine, also dish soap/wash-up liquid or liquid hand cleanser can be used, it doesn't make that much difference with coarse-bristled brushes and makes zero difference with synthetic brushes.
Repeat the washing if necessary.
Once you've thoroughly washed and rinsed the brush the bristles should be squeaky clean. Re-form the brush head and put it aside to dry. Flat, hanging tip-down or in a jar with the bristles pointing up generally makes no difference any more, but if you want to err on the side of caution hang them or dry flat.
Phew! You're probably tired just reading that, any wonder nobody wants to do this properly on each and every brush used every single time? And it's no joke, cleaning a few brushes really well this way can take the guts of an hour.
Note: when using the brush for solvent-based paint the brush should be absolutely dry before you use it again and overnight drying is often not enough. It's fine to use a slightly damp brush again if using a waterbased finish.
You can clean a brush perfectly well using solvent only. The reason most don't do this is because of the amount of solvent required, and the space the containers take. Re. the note above in the traditional method about the amount of solvent, for this method to work well you'll need at least a couple of litres and again, more would not hurt.
For oil-based varnish I would use at minimum two rinsing containers, but three or more is better. Some people use as many as five and report that it does make a difference. Personally I doubt if you're doing it right that more than three is necessary, but a fourth could be added for security.
- Wipe the brush dry first.
- Rinse in container one and dry out the brush as described above.
- Now do at least two rinse cycles in the second container.
- Dry again and rinse well in the third container.
- Dry off the brush and re-form it into shape and put it aside to dry.
This time it's okay if the brush is still slightly damp when you go to use it again as it's damp with the right solvent, and you'll probably want to dampen it before use anyway (see related point at bottom below the separator).
To monitor how well you're rinsing out the brush the solvent in the third container should remain clear, without any tinge of yellow, nearly indefinitely (literally years if you're not cleaning brushes on a regular basis). If it does start to become yellowed it is not the end of the world, but it indicates you should be being a little more scrupulous in the preceding steps or that you're not using a sufficient volume of solvent.
If you conscientiously use either of the above methods well your brushes will last and last — I have numerous brushes that are over ten years old and still going strong. But that's nothing, many professional finishers and painters report that they have brushes that are 20-50 years old o_O
Preloading the brush with solvent
The tip given in the answer by LeeG of moistening the brush prior to loading it with the paint or shellac/varnish is one given in a few good guides to finishing. The primary reason to do this is that it's an aid in smooth brushing, it helps the finish flow out from the brush (and it is excellent at this). A second reason that might be given is that it helps with rinsing out the brush afterwards (note that the longer you paint the less help this is, particularly with fast-drying finishes).
Some guides may give both or just one of the above reasons, and may go on to state that it helps prevent the paint from wicking up into the base of the bristles inside the ferrule but they're completely wrong on this, it actually has the reverse effect in practice. This should be perfectly logical, thin liquids wick much better than thick liquids. But if anyone has doubts try comparison tests and see for yourself.
The best way to help prevent finish getting up into the ferrule is to never load the brush more than halfway up the bristles.