Well if you want to join the Royal Navy in the 18th century you're in luck, apart from the fact that logistical planners estimated around 50% of the sailors on any given voyage were likely to die on said voyage. In a Peacetime year around 50-70 thousand men would join the Royal Navy and in a Wartime year it could be around twice that.
So why are you in luck? well, because Impressment was in force at the time. The Impress colloquially known as "Press Ganging" was a method used by the Navy (The Army briefly experimented with trying out press gangs betwen 1778 and 1780) to find crews for vessels, by forcing into service "eligible men of seafaring habits between the ages of 18 and 55 years". So all you really had to do to get yourself a commission was get caught serving on or even just seeming to serve, have served or be capable of serving on a merchant or fishing vessel by press gangers roaming the docks hunting for eligible young men. Although the press gangers were primarily looking for seafarers with skills and experience sailing there were occassionally non sailors with absolutely no seafaring skills or experience picked up by press gangs anyway presumably when deadlines were tight and quotas needed to be met so they resorted to just grabbing any random able bodied man they found around the docks that day.
Records submitted to parliament for the period of 1775-77 stated that 24% of sailors recruited were press ganged into service, with volunteers presented as two separate groups of 47% and 29%, (other records from other years were consistently similar) it is not entirely clear why the volunteers are separated into two groups, but it is estimated to be likely that one of the groups is the true volunteers while the other is the "volunteers" who were being press ganged and decided to volunteer at that point. It was a bit of a tough choice to be honest with you. A volunteer got a nice signup bonus while someone pressed into service did not. However a volunteer who deserts their vessel is executed if caught, while someone pressed into service is merely escorted back to their ship and forced to continue their service. I cannot tell you with any confidence which of those two 47% and 29% groups of volunteers was the true volunteers and which was the "volunteers". Honestly though if I had to guess the larger portion may actually have been the true volunteers. As hard as Navy service was, I previously mentioned non seafarers were rarely pressed, and for someone who was already accustomed to a hard life at sea, service on a Navy vessel was generally considered easier than service on a Non Military British Ship (especially during peacetime) for a few reasons, namely that the crew was much larger, around 4 times the size, but unless you were actually getting into a scrap there weren't that many more jobs to do so the individual workload of manual labour for each Sailor to bear tended to be much lower. Combining this with the fact that the pay was a little bit better (Though they routinely held your wages for an incredibly long time as a measure of leverage to encourage against desertion) and the food was a lot better. You may not get rich as a Navy's man, you may die of disease or combat, but you were fed much better than anyone else was, both in quantity and quality.
You didn't actually "Join the Navy" as we understand it today. Whether you came by force or by choice your commission was with the Vessel and for the Voyage. When that commission with that Vessel was up, you were no longer a Navy's Man. You would be encouraged to volunteer for additional service and seek another commission transfer to another detail of course, and may end up finding yourself more than merely "encouraged" later on down the line when some other vessel is seeking to employ press gangs to fill its crew for their upcoming voyage but you had no formal obligations to serve in the Navy as a whole outside of your terms of service to the vessel that held your commission. The law was only amended to grant someone who had been pressed into service exemption from further impressment once they had finished that service in 1835 so for the 18th Century you could easily find yourself forced into the Navy, serve your commission until you are released and make it home only to then get forced right back onto another Vessel if you were unlucky. (Or just had one of those faces I guess.)
I’ll admit I’m not the most qualified person here to answer this, but I’ll say it depends on which arm of service you are aiming to join. Since someone has already covered the Royal Navy, I'll describe the process for joining one of the Royal Army’s Regiments of Foot- the infantry. While it changed somewhat over the course of the 18th century, it remained broadly similar.
In this case, one would join by encountering or seeking out a recruiting party. The party would typically consist of a junior officer (Lieutenant, probably), a sergeant and a corporal, and a drummer and/or fifer, all from the same regiment. Recruiting parties would move around the United Kingdom throughout this period and into the early 19th century, going between major cities, small villages and especially hitting market fairs. Often, they'd set up shop for a few days at a particular location (most often a tavern or pub), parading through the surrounding area calling for recruits. The party would hang around the tavern as well, talking to locals, buying them drinks and trying to persuade them to join them (often with a healthy dose of alcohol involved). In fact, the story of the soldier who joined the army while drunk became a staple of British folk songs and art.
If someone was interested in joining (called "enlisting" in this period), they would go before a local magistrate or Justice of the Peace to attest they were joining the regiment, and if an apprentice or servant were doing so with their master's permission and were healthy and of sound mind. The recruits were supposed to be over 5'6" tall, and between the ages of 17 and 45. Officers were often instructed to avoid recruiting those who had had hernias or back injuries or had epilepsy (These standards were often relaxed during a war…). They then would then be given the “King’s Shilling”- a portion of their first month’s pay and a recruiting bounty. Part of this would immediately be taken back as a part of their pay was used to pay for their uniform, but it could be a few shillings if there was a war on and recruits were scarce. At this point, they probably wouldn’t be issued more than a cockade to put in their hats.
Depending on the period in the 18th century, the recruits might either be sent along a central training depot (often when the regiment they joined was stationed overseas), or straight to regimental headquarters. There, the recruit would receive a medical examination and be officially placed in the regiment (If they DIDN’T pass the medical examination, the officer of the recruiting party was on the hook for the money already paid. At this point in history, the individual regiments of the army were largely responsible for their own recruiting. So, if a recruit met a recruiting party from the 42nd Regiment, The Black Watch, and joined them, he’d be joining the 42nd Regiment. The recruit would be trained and equipped at the regiment, unless the regiment was overseas, in which case, they’d start getting their uniform and some training before shipping out.
A good source for this is Cuthberston’s manual for officers:
Cuthbertson's System for the Complete Management of a Battalion of Infantry
https://play.google.com/books/reader?id=1SxEAAAAYAAJ&pg=GBS.PA66&hl=en