First off I am history enthusiast and have no proper education on this specific subject, so take what I say with a few healthy pinches of salt. Hopefully someone more educated and with more insight can come along soon and give better informed answer. Most of my knowledge of this comes from having grown up in a county that exported ice back in the day, with my grandpa recounting my great-great grandpa's work in the field, and just generally reading up on it over the years. I will provide sources in English where I can, but most the sources I know are in Norwegian.
First of all, I focus on the ice trading of the 1800s and later, as that is what I know the most about, and to my knowledge there aren't that many sources for most ice trade before that. That said, many of the techniques used in the 1800s could easily have been used earlier, if on smaller scales. For reference (Norwegian source), in 1898 Norway exported ~554 000 register tons, or ~1.4 million cubic meters, of ice. From my understanding that likely accounted for something around 40% of the global export market on ice.
Either way, the modern ice trade of the 1800s is generally attributed to have started in the US and quickly spread from there to other viable regions. In most cases this involved making an ice dam, which could be as 'simple' as stacking rocks in a stable wall that is about 6 feet thick and a desired height, usually around 6 feet as well. This dam was build around a relatively small indent in the terrain where water naturally gathered, such as a bog. Once build it would collect water and during the fall a small group of workers, usually the owner and their family, would regularly clean the water of debris using nets or whatever else they deemed useful. Clear ice was critical for structural integrity which was important for handling and transport, but also sales reasons, as buyers naturally would prefer clean ice without leaves and twigs. As a side note, freezing fish in ice for long transport journeys were also not unheard of in the 1840s (Norwegian source).
Once the ice hit a safe density to be walked one, 10 cm or more, the more labour intensive parts could start. Again for both structural integrity and sales purposes, the ice had to be worked to reach the congelation ice layer, called "stålis", or steel ice, in Norwegian. In layman terms this is just clear ice. This ice is what forms under the top layers of ice and can only be formed when water is not in motion (relatively speaking). It has a lot higher density and thus contains a lot more cold, and is easier to handle. Reaching said ice involved sweeping snow off the ice, and in very thin layers scraping off the top layers of ice until you hit the clear ice.
This was done until the ice was roughly between 30 and 60 cm thick (1-2 feet) depending on what the standard shipping size was with whoever you worked with. For transport to England a 2x2x2 feet cube was supposedly the standard. When the desired thickness was achived the ice was cut and either transported to an ice house (insulated storage) or directly to a waiting ship. This was initially done by sliding the ice on a run build of wood (think a square bobsled track with people pulling and pushing ice), though as infrastructure modernized, especially big places got a dedicated train track.
Before we get on to the transport of ice, I think it is important to mention that well stored ice, could last for up to a year in European climates. This relied heavily on ice houses, which were heavily insulated buildings of various designs that could keep a very cool and constant temperature through out the year. The Yakhchāl is perhaps the most interesting, being an evaporative cooling building, which allowed for ice to be stored long term even in warmer climates such as Persia.
Other insulation techniques involved covering the ice in things like saw dust or hay which provides insulation as well. This type of insulation was used a lot in transport.
Generally there is a significant lack of documentation of how the ice trade worked in terms of sales and contracts and other practical legal and economic matters, at least within Norway. It is clear however that contracts seems to be a norm (Norwegian source). One contract mentioned in this source is from an English company purchasing between 200-300 tons of ice for the upcoming winter. It contains a clause roughly stating "if the Norwegian weather allows", removing liability for the distributor and ice producer in case of a bad winter.
No matter which way the sale was determined, by contract or otherwise, the ice would be loaded on a ship insulated by hay and saw dust. In most cases the ice was actually still transported by sailing ships all the way up until the 1900s, at least from Norway (Norwegian source). This was mostly an economics matter, as older sailing vessels were relatively cheap to aquire and operate compared to steam power. The ship would then head to its destination and deliver or sell the ice there and often come back with other goods to be sold in Norway.
The ice was then stored in ice houses or ice cellars at the destination and brought out as needed to fuel ice cabinets or other things. Most of the ice from Norway went to the UK and France, but also the Baltic region and the Mediterranean. Supposedly in some extreme cases it went as far as Congo, the West Indies, and South-East Asia.
In Norway the practice of ice harvesting presisted all the way up until 1968, though it in general was in a sharp decline after WWI (Norwegian source).
I have primarily used this (Norwegian source) publication as my source as it is the most recent one I have read, and concurs most with what I have read and heard over the years.
This page (Norwegian source) has a good video showing a slide show of some workers in the various stages of working the ice.
This article provides a good overview of a, at the time, innovative natural ice 'factory' in Germany at the time.
Lastly Before the Refrigerator by Jonathan Rees, is a book about the American ice trade, you might want to check out, though I have not read it and cannot speak to its quality.