Generally, what's done with your body by other people depends on where you live and how quickly you are found. We'll assume the most common case scenario, and say that you're found in less than 24 hours and you haven't written a formal will that details any specific funerary arrangements.
Rural Areas
If you live in a secluded commune or a sort of village, you're very likely to simply be given a quick washing and then buried whole into the ground, where your body will be used as nutrients by the vegetation and fauna, and won't cause any of the typical problems to others such as foul scent or emotional shock. Most communities will customarily mark your burial spot somehow, be it through a grave, altar or memorial. Depending on the size of the community, there may not even be a dedicated mortuary workforce, and local mayoral workers will handle your paperwork and produce a death certificate for you, before handing any legal rights over to your closest living relatives (for minors and unmarried bachelors, the rights holder is usually a parent/guardian, and for married adults, it's their spouse or one of their adult children). The person who holds these rights is called a personal representative, but informally, terms like "next of kin", "executor" and "administrator" are also used.
Urban Areas
If you live in a city or town, bureaucracy and authorities get involved, and the process becomes more elaborate and complicated.
Initial Discovery
Whoever finds you will be expected to call emergency services or their local police station, and report their discovery. Police officers will arrive to secure and evacuate the general area; depending on how the state of your body is described, there might also be a team of paramedics sent over to attempt resuscitation and, if nothing else, at least confirm your death.
Preliminary Assessment
Following that, a coroner will be notified of the case and will also arrive, sometimes accompanied by a team of forensic investigators (it heavily depends on the resources and funding that's available to their department). A preliminary examination will be made on your body, which assesses the general damages and helps forms theories about the likely cause of death, and then you'll be stuffed into a biohazard body bag and driven to a morgue, where you'll be placed inside a freezer to prevent your decomposition.
Meanwhile, the forensics team will harvest samples from your body for analysis, while the police will temporarily seize some of your items (ESPECIALLY any electronics you have, e.g. a computer or phone) and do a background check on you to form a profile of your identity. They'll gather as much information as possible on you and open an investigation into your death, to assess the exact circumstances behind it and, most importantly, make sure that it really was a suicide, and not a murder or assisted suicide. Your seized belongings will be returned to your relatives only after their investigation is closed. This investigation is an entire process of its own, but that's not what we're here to talk about...
Morgue Admission
At the morgue, the coroner will make various phone calls to people that you know (or, if this is unknown, people that you might know) in order to inform them of your passing if they are not already aware, and establish contact with your closest living relative, whom will be periodically updated on the status of your body with any new information that comes to light.
Then, your body will be put on a waiting list for autopsy (a surgical process that aims to scientifically determine a body's exact cause and time of death). Depending on the volume of bodies and the number of staff that the morgue possesses, it might take them anywhere from months, weeks or days to get around to your body, but the freezer will keep it in good condition.
Not all bodies need to be autopsied, especially elderly bodies and those who die in hospice care, but suicides typically present fairly mysterious and potentially even high-stakes circumstances that necessitate it.
Autopsy Time!
Once it's time for your autopsy, the coroner may deem it necessary to perform various medical processes that I will not outline here, both for brevity's sake and for the fact that there are so many possibilities to go over. Still, you will likely be cut into in order to have your internal structure more closely examined, and then sewn back up with sutures. A report will be made on your death that describes the physiological reason you died with maximal precision, as well as any additional notes that the coroner deems necessary to include (this includes the fact that it was a suicide.).
Meanwhile, a mortician will handle any paperwork and documents that need to be created in your name, including your death certificate. Some of these will be kept by the state, while others will be released to your next of kin. Your body is then discharged from the morgue.
Funeral Arrangements
Your body is then relocated to a funeral home (funeral homes are distinct from morgues and serve different functions, though it's not particularly uncommon for the two to exist very closely together and even share much of the same staff.), and your next of kin will be summoned to discuss funeral arrangements. Do they want a funeral? Do they want you cremated (reduced to ashes and preserved inside a container)? Embalmed (preserved with formaldehyde in preparation for a funeral)? Closed casket? Open casket (where the body is restored to a peaceful appearance by a mortuary artist and visually displayed)? Picture viewing? No viewing? This is for them to decide.
If you are a legal adult, it's possible to make your own funeral arrangements before you die, and generally, I'd say that it is a good idea. It removes a lot of financial strain from your loved ones and simplifies the process for them, allowing them to better grieve. On top of this, it gives you more agency over what exactly is done to your body, just as you would want it. Plus, you get to write something really stupid and funny on your gravestone.
In Memoriam
Finally, we reach the part that most laypeople are familiar with. If you were cremated, your next of kin will simply get to keep the container (usually a vase or an urn) containing your ashes and do as they like with it. There may not even be a funeral if they do not opt for one. If you were embalmed, the funeral ceremony will be held, weeping will be had, and after it's done, you'll be relocated to a graveyard, where your body will spend out the rest of its days six feet under, typically within a casket or a tomb.
You may be visited by relatives, or you may not. Someone may pass your gravestone and briefly ponder and speculate about your life, considering if they may have gotten to know you in another time... and then move on and forget about you. Eventually, your name will be muttered for the last time, the last memory of you will be had, and then you will have truly experienced your final death.