How do I sign up? Can I, considering I have no experience at sea? Will I work on a military vessel? How will it be assigned and how do I learn my trade? How do I learn new skills and rise in rank?
Bear in mind that my knowledge concerns the Georgian navy, so about 50 years earlier than your inquiry. However, I think that the answer will still have some value. If you were between the ages of 18 and 25 you would not have been joining the navy at all. It was the custom for people to go to sea at a very young age indeed, as early as six or eight in some cases, normally as a "servant" of some sort, which kind of meant an apprentice. You might be unattached and used as a powder boy or have other duties. One of the big uses of children and teens in the navy was as the people tasked with reefing the sails at the top of the masts, where their agility was most useful. Most officers got their start as the servant of an officer and were meant to complete two years as a seaman before being promoted. If you were going to be a boatswain or a carpenter or anything else, odds are you came on board as the servant of a boatswain or carpenter, probably through family connection of some sort and would be noted as such on the ships roll. Often these "boys" would make up 10% or so of the ships company and would be a mix of young gentlemen on their way to becoming officers and poor boys. It would be possible, if you showed yourself to be horrendously talented to become an officer, either early or late in your career, around 10% of lieutenants and above were of non gentlemanly origin. You would have had the choice of weather you wanted to work on a naval vessel or a merchant one, generally the work was easier on naval vessels, there was a much lower ratio of men per tonne on a naval vessel. The pay was better, as was the food and there was the prospect of loot to be gained from enemy ships. On the other hand, you were slightly less likely to die on a merchant vessel, although the possibility of being wrecked as just as high. That said, the door between merchant and naval service was a revolving one.
It is important to note that people did not join the navy as we would today. It was not an institution in the same way as today. People joined a ship and were in practice a member of that ships company, not the navy, until he was discharged. It functioned more as a collection of ships than a coherent institution. Normally promotion came through patronage and ability. If you were a good ordinary seaman you might be promoted to boatswain on your ship. If you performed well your captain/admiral might recommend you to another captain for service on his ship and you could accept or decline as you wished. Indeed, it would not unusual for a captain, when he changed ship to take his entire crew with him to the new ship. The loss of the patron could be devastating to a rating's career, if a captain died he would be left without protection or opportunities in an unfriendly world.
"The Wooden World", N. Rodger
William Robinson's Jack Nastyface: Memoirs of an English Seaman recounts how volunteered to serve in the RN in 1805. He doesn't fully explain how he enlisted for service, but starts his memoir when he sent aboard a receiving ship. He is examined by a doctor who deems him seaworthy and is ordered down to the ship's hold for the night.
The next day he is ordered to the admiral's tender(generally a small fast ship used to ferry things/people between the shore and wherever the admiral's ship may be). Along with William, "...nearly two hundred, including a number of the Lord Mayor's Men, a term given to those who enter to relieve themselves from public disgrace, and who are sent on board by any of the city magistrates. These poor fellow have a sad time of it, as they are the derision of the old and more experienced and hardened sailors, who generally cut the tails from their coats, and otherwise abuse and ridicule them."
Once on board the tender all the new recruits are sent down to the ship's hold and locked in and guarded by marines, "...with their muskets loaded and fixed bayonets, as though we had been culprits of the first degree, or capital convicts."
While traveling locked in the stuffy rat infested hold, many of the men become seasick, vomit, some faint, and only when an officer notices their plight are they allowed fresh air. William arrives at the Nore where he is soon taken to his assigned ship(he doesn't name the ship). Before arriving at his ship, he receives purser's slops, cheap clothing that the price of which is deducted from his meager pay.
Now he is taken to his ship and encounters what he called the birds of passage. A system of plunder where new sailors would have their shoes or blankets stolen in the night. He is next drafted to a ship-of-the-line which was being readied to join Nelson's fleet.
He doesn't go into anymore detail about the basics of naval life, he talks more about the actions he fought in and places he traveled.
To answer your question about signing up without experiences, the answer is yes you can 9 times out of 10. The navy was in dire need of men thus why press gangs are still so well known. Additionally, for both the Army and Navy, taking on criminals was a simple solution.
The only time a inexperienced man might be turned away would be from a popular ship(often a frigate) with a popular and successful captain. Some captains would put up handbills or posters advertising their ship. A good example is Lord Cochrane's poster stating, "The Flying Pallas Of 36 Guns At Plymouth, is a new and uncommonly fine Frigate. None need apply, but SEAMEN, or Stout Hands, able to rouse about the Field Pieces, and carry an hundred weight of PEWTER, without stopping, at least three Miles."
Inexperienced men were rated as Landman earning 1 pound, 2 schillings, 3 pence per a month. Learning seamanship was basically an apprenticeship. You were assigned a division, often the Afterguard, which would have you working the lower sails with ordinary seamen. Landsmen would often start off by doing simple physical tasks like hauling rope or working the capstan. Men doing unskilled work were called waisters, as they were in the waist of the ship, and were the lowest of all the crew.
After a year or two of service the Captain, if he deemed your seamanship skills fitting, could rate you as an Ordinary Seaman(1.5.6 per month). With more experience you could be further rated as Able Seaman(1.13.6 per month). From here there are a lot of different positions to fill. You might become skilled enough to become a boatswain's mate, or a carpenter's mate, a man of the sheets, forecastle man, or captain of the tops, etc.