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So you're a British sailor onboard a merchant ship in the year 1720 and you get attacked by pirates. First of all, the attack could happen in a number of different ways. The most common way would be for the pirates to attempt to approach your vessel to within hailing distance while flying the flags of a friendly nation. Pirates certainly did not just sail around the ocean flying the "Jolly Roger" as that would give them away and every ship would be able to avoid them from a great distance. However, no other ship would have flown their colors all the time either since flags were expensive and they faded and wore out with prolonged exposure to the elements. All ships normally only flew their colors for a good reason. When a ship was sighted on the horizon, sailors would not necessarily know which nation it belonged to, although they could sometimes guess from the design. Even at closer ranges in calm weather when colors were customarily hoisted, it was extremely common practice to hoist false colors to deceive other ships, and not just for pirates to do this but everyone. Most ships would have a variety of different national flags onboard incase they needed them (in the case of pirates they might have captured these from other ships). If there was war between Britain and France, a British ship might hoist French colors if they thought they were approaching a French ship and vice versa if a French ship thought they were approaching a British one. What this all meant is that colors were almost worthless as a means of identifying ships at sea and no one trusted them. However, this all depended on context as well. Piracy was not ubiquitous at this time so unless a merchant ship had reason to suspect there were pirates around because they had heard reports of them, they might be more trusting. In fact, until the late 18th century (after the heyday of piracy) ships would often have incentive to hail and communicate with other ships they encountered at sea in order to compare longitude estimates because the marine chronometer hadn't been invented yet which made longitude estimates very crude. Another reason would be to exchange valuable news, such as declarations of war or reports of pirates.
So taking all this into account, what would a British merchant ship actually do when they sighted another ship? They would have three options: avoid, approach or do nothing and continue sailing its course. These courses of action would often depend on the actions of the sighted ship and could change at any time. Sometimes pirate ships would immediately unfurl all sail and give chase after sighting another ship. This was understandably suspicious -- it would be like if a stranger suddenly started aggressively running toward you with a mean look on their face. Most merchant ships would flee in response and a chase would ensue. Other times pirates would "heave to" which meant remaining stationary and they would signal for the other ship to communicate. Seeming overly eager to communicate like this could also be suspicious also. Another tactic was to signal for help as if in distress, but this too was usually seen as very suspicious and ships might not even approach a real ship in distress for fear it was a pirate. More subtle tactics included disguising the type of vessel to appear more like a typical merchantman by shortening the mast and hiding most of the crew below decks, and to follow the ship in a less direct and obvious way. Still, ship captains weren't stupid and these ruses often didn't work.
So let's say our British merchantman gets suspicious and decides to sail in the opposite direction of the mysterious ship they have sighted because they know there are pirates around and it's acting strangely. Now the pirate ship starts following them and unfurling its sails to gain extra speed. A chase ensues. Sometimes if a pirate ship was small and it was a large merchant ship, they would stand and fight or even sail toward the suspicious ship to give the impression of strength and confidence and this could seriously intimidate pirates who suspected it was a navy ship or who just weren't prepared to put their lives at so much risk for a chance of plunder. Unlike regular soldiers, pirates are solely motivated by a desire for monetary gain so if a ship put up determined resistance or seemed like it was going to they would often give up and search for easier prey, though this wasn't always the case especially if the ship was thought to be carrying a lot of wealth.
The typical course of action for a merchantman would be to flee and attempt to avoid a fight altogether if they thought they could, but it was very common to quickly give up and surrender if the pirate ship was close and obviously faster. If it did come to a chase at sea, it was a complicated affair involving all sorts of sailing knowledge and judgements about the wind, but one tactic they could use was throwing things overboard to gain speed by lightening the ship. Another tactic was unfurling all sails to a dangerous degree in an attempt to gain more speed which could cause the mast to break under too much pressure from the wind (think of it like running too fast down a hill and tripping) -- if this happened, the ship would be captured or have to fight. Another desperate tactic for evading pirates if the ship was close to land and especially if it was a small ship was to deliberately run thee ship aground somewhere from which it would be much easier for a small crew to defend since the pirates would have to send out a longboat in order to take it. Alternatively, the merchant crew might just run off onto whatever island or shoreline they landed on to escape the pirates and burn their ship to prevent the pirates from capturing it. This was obviously a last resort. Pursuits could also last for several days or longer and at night it would be much easier for a merchantman to slip away and lose the pirate ship by altering course. When chasing at night both the pirate and prey would normally extinguish all lights to hide themselves, but sometimes the prey would deliberately keep a light going and then set a false fire in a boat or tub on the water hoping for the pirates to be fooled and continue heading toward that while they changed course.
If the pirates caught up with the merchantman, there were a few things that could happen. Once in hailing distance (about half a musket shot or a hundred yards) or when the ship clearly knew they were a pirate, the pirates would usually raise their flag or "Jolly Roger" which could be black with a skull and crossbones or a skeleton holding an hour glass or it could even be entirely red or red with an arm holding a sword. Either way, the message was to surrender or die and when the merchant ship was within hailing distance they would call out through a speaking trumpet across the water for them to lower their topsails (thus reducing their speed as a sign of submission) and surrender or not be given any quarter when captured. This usually intimidated most crews into surrendering but not always.
Usually the master of a vessel or the captain would have much more of a vested stake in making sure the ship's cargo was protected and got safely to its destination while the crew wouldn't. The crew might not be paid if the ship was captured, but that was usually judged better than risking death. You can imagine there would have been a lot of serious and heated discussions aboard merchant ships in this situation and sometimes the crew would just insist that they surrender even if the captain wanted to fight. If they didn't surrender either by refusing to lower their topsails and continuing to try to escape or by preparing to fight and running out their cannons, they would be fired on by the pirates.
Follow up question, let's say I, the sailor in question, am interested in joining the pirates, and upon being attacked I successfully join their crew. How would I be treated relative to the rest of the pirate crew? Would I be treated like scum and be expected to prove myself, or immediately treated as equal? Or, perhaps more likely, something in the middle? Might my behavior during the attack make a difference?
Ideally, are there any first-hand accounts of sailors joining the pirates I could read? (merchant or navy sailors also). Even more ideally, are there any surviving primary accounts of sailors joining the ranks of the more major pirates like Blackbeard, Vane, Bellamy, etc.?