One of the things that of home brewers will figure out rather quickly is that clearing beer is not as easy as following the advice on the internet. If it were, commercial breweries wouldn't own filters, which take up a massive footprint, are costly and introduce a potential source of oxygen and iron contamination.
Before we begin, lets talk about the various types of advice, the various causes of haze and then the types of filters that are available commercially and for the home brew scale.
Cold crashing is a way of causing your yeast to go into a sort of hibernation mode. The thing is, this only works, and even then, only partially, for ale yeast. Yeast contain FLO genes and the more copies of them, the more likely they are to flocculate. Temperature will influence this, yes, but more importantly is the presence of maltose. Maltose "clogs" the arms on yeast that will link up and make yeast sink to the bottom. Fewer FLO genes mean fewer "arms" which means smaller clumps. More arms means that yeast can start to floc out even if there is residual maltose, leading to an under attenuated beer. In general, yeast that floc out sooner have a lower attenuation rate and yeast that floc out slower have a higher attenuation rate. Cold crashing will only influence this partially.
Adding something to cause haze to precipitate out depends on two things: 1) what you're adding and 2) what's causing the haze. Haze can be caused by yeast, protein, glucans or polyphenols. Three out of four of these will respond, at least partially, to adding something. Glucans, or haze formed by abusing the mash or over sparging, will never precipitate out and filtering will often make it worse. It also seems to be the most common type of haze caused by home brewers who don't necessarily have the equipment available to gently transfer mash, or who need to stir quickly and vigorously to correct a temperature error.
There are many products that can help, and they all work using the same principle, namely Stokes Law. Stokes Law, in brief, is a calculation for the speed at which a denser and larger particle will fall through a liquid medium. Additions, either kettle finings or post kettle finings, work by attracting particles and making them bigger. Collagen-based finings, such as isinglass or gelatin work on yeast and to a lesser extent on protein. Carageenan works on protein and plastic-based finings work on polyphenols, though they're usually still too fine to precipitate out and require filtration.
Commercial filters come in various shapes and sizes, but can be roughly divided into two types - counterflow and depth filtration. Counterflow filters rely on the flow of beer to clear away anything that might "blind" the filter. The filter itself is extremely fine and is measured in units smaller than microns. These filters are extremely expensive, but have the advantage of having filters that only occasionally need to be replaced and no need for a filter pre-coat.
Depth filtration is less expensive and often requires disposable filters as well as a substance that builds up against the filter to provide a much finer filtration. That substance is usually kielselguhr, which, if not from a reputable supplier, can contaminate your beer with iron. It also requires careful premixing with deoxygenated water or it can oxidise the beer.
Both of these filters are usually only used as a polishing and for extending shelf life.
On the home enthusiast side, you have the options of plate-and-frame or canister. Neither of these are designed for either counterflow use or depth filtration. They are made of woven cellulose, which is why it can make a glucan-caused haze worse.
Canister filters are easily available and often sold as "water filters". There are various grades of filters available, often with an absolute micron rating. There are a couple of problems with them, however. The filters themselves have a plastic housing and can't be re-used. I've seen "re-useable" canister filters, but they're designed for water and you're inviting infection if you use them for beer. They also don't really work very well beyond rough filtration. I've never had a brilliant beer come out of one, or even two in sequence. They tend to blind very quickly and require a substantial amount of pressure that isn't held very well.
Most brewers avoid plate-and-frame filters because the only model they're aware of is the Buon Vino Mini Jet, which is designed for wine and has the obvious flaw of being open, meaning that oxidation is impossible to prevent. That's fine for wine, which requires a small amount of oxidation for aging, but not for beer. Instead, I recommend the Vintner's Shop sealable filter. It runs entirely on pressure like a canister filter and the filter pads don't have a plastic housing. It's the only filter that I recommend as it's the only home version of a filter that actually works - as long as you're using the #4, or polishing filter pads.
In general, you shouldn't be filtering your beer at the home brew level, but occasionally there are those lagers that you want brilliant and you want to remove the muddiness that an unpolished beer can have if it's something like a Helles or a Pilsner. In that case, there's only one filter I recommend. Even though it may be hard to find it in stock, you're wasting your money if you buy anything else.