I have linked to the relevant wiki pages for further reading. I have marked the TLDR keywords bold.
Wheelbarrows generally slow hauling down. Only 1-3 dwarves can haul, and they have to go get the wheelbarrow before they can walk towards the actual item. However, wheelbarrows negate item weight, which means very heavy items can be hauled faster. Thus, I recommend to only use wheelbarrows for hauling heavy
stones and (filled)
animal cages.
Barrels can contain several foodstuffs (including bags of seeds). Similarly,
bins can contain several small items (including raw materials and finished goods). Both are useful because they fit a lot of items into small stockpiles and make gathering faster (because a single dwarf will pick up several items). However, the bin/barrel will be "locked" while a dwarf tries to insert or take items (even if they are still miles away). Thus, I recommend them whenever you want to
save space and have
few dwarves using them. For example, leather comes in small stacks, and it usually gets produced by only one tannery and consumed by only one leatherworks. Therefore leather stockpiles are ideal candidates for bins. An example for when
not to use bins, cups/mugs/goblets are frequently used by every dwarf to drink, so bins would be a bottleneck, lead to cancellation spam, and generally frustration.
Special caution must be taken when using the
Quantum Stockpile exploit. Read the wiki for a full list of potential problems. If you want to exploit technical details, you must pay attention to technical details.