It is unfortunate that these terms have such loose definitions these days that it is very hard particularly if you are a writer to try and paint a picture using these in your work. I know. I am a writer. I define them like this:
Alchemy: Alchemy is the forefather to modern chemistry and it was primarily concerned with converting metals from one type to another. Particularly turning lead to gold. Ancient alchemists sensed there was a relationship between the two since their weights were very similar in nearly equal amounts. On the periodic table you can find gold (AU) is very close in atomic weight and atomic mass to lead (PB) so they had the right idea but without advanced atomic technology, they would never be able to make their dream a reality. However, it did not stop them from trying to create magical sigils, images, and processes they believed would do the trick. Somewhere along the way, they began to get predictable results and chemistry was born.
Sorcery: Is one form or name for the creation or manipulation of magical events using rituals, materials and hidden or secret knowledge. The word's literal meaning translates to "one who influences fate." Sorcery has different meanings to different people so it can be difficult to lock down completely, even among writers. Michael Moorcock for example defined sorcery as the ability to attempt to compel or control demonic forces in his Elric/Stormbringer series of works. Other writers such as Steven Brust, define sorcery as the scientific application of magical energies through the Imperial Orb, and artifact within his world of the Jhereg series.
So, it will very widely depending on who you ask. The safest course is to assume it is a means of manipulating reality using complex, hidden rituals, which may use hand signs, bodily postures, exotic materials and information from potentially supernatural beings such as demons, angels, imps or devils. It is commonly part of the lore that sorcery is not considered a favorable practice and is considered a "dark art." This interpretation may vary widely from culture to culture and most human cultures on Earth discuss different types of magical traditions with the most well known being that of the Hermetic Mages, which are the most common origin for magic as it is known in most Western cultures. This is a very complex topic and you can find an incredible amount of information about it in your public library. I have given you just the tip of a very large iceberg.
Witchcraft: Another term which varies widely with the culture discussing it, as well as the storytellers relaying the tales. Generally speaking, when you talk of witchcraft, it is related to the pagan rituals of worshipers of the Gaia force or Mother Nature. It is a non-denominational religious set of beliefs surrounding the worship and adoration of natural phenomena, also known as Wicca. If you are looking to do further research, that would be your preferred starting term.
As with other words in regard to magic and writers, witchcraft resembles sorcery in its use of incantation, physical props including wands, staves and rods, natural herbs, weather-related phenomena and the creation of minor enchantments and spell use. However, books such as JK Rowlings, Harry Potter series, includes a wide array of magical use under the heading of witchcraft.