The method of chlorination to use depends on part on the suface of your pool, the type of filter you use, and your geographic location. For example, stabilzied chlorine (trichlor/dichlor) works well in areas with a short swim season that winterize pools, especially with sand filter, since the CYA is constanly being diluted but are a disaster with cartridge filters or non backwash DE filters in areas with extended swim seasons that keep the pools open year round.
Trichlor is a good choice on NEW plaster pools because it helps keep the pH in line as the plaster cures but you do need to keep a watch on the CYA levels, particularly if you don't have a sand filter.
Metals (call them minerals or ions if it makes you feel better but they are metals--copper, silver, and/or zinc) are NOT primary sanitizers but only supplimental ones that MUST be used with a residual sanitizer (usually chlorine). In theory they will allow you to use a lower FC level but in actual practice this is not advisable.
The type of chlorination that is used might change depending on the time of year, the age of the pool, and also if any equipment is changed. There are basically only 2 types of clorine--stabilized and unstabilized, and which one is a better choice depends on the factors I listed above.