It depends a lot on the size of the ship and the type of dock it might be at. Small ships, such as say sloops or brigs, might actually have their own sweeps (oars) and could boom off the dock and sweep down the harbor. Larger ships might be towed by the ship's boats, or by using their own boats to take an anchor out, drop it, and use the ship's capstan to pull it up to an anchor (this process was called "warping"). The goal would be to get the ship far enough out into the harbor to be able to lower its sails and catch a breeze, then work its way out of a harbor.

For a lot of use cases, the winds that prevail to get a ship into harbor will block ships from leaving, so ships would have to wait for favorable winds or have a multi-step process to leave a harbor. I explained this in some more detail in this comment, to quote from that:

Because these ships were, after all, sailing ships, the duty of the fleet becomes more difficult because of the winds and current conditions that could be experienced in a particular area. Broadly speaking, winds that would allow for ships to leave a port would tend to blow the blockading fleet offshore, while winds that kept ships in a port would blow the blockading fleet onshore (which is one reason why clumsier ships would be kept offshore, so as not to be wrecked). Obviously, close attention to the weather and watching out for storms was a major responsibility of ships on the blockading fleet. Additionally, blockading fleets still used up the same amounts of victuals (food, water, etc.) and naval stores (sails, spars, cordage, tar, gunpowder and shot for practice, etc.) as a fleet under sail would, so plans for supplying the fleet were crucial. Most admirals attempted to keep enough ships on station so that one or two could always be rotating back to a friendly port to re-provision and bring out mail and news.

Looking specifically at Brest, the dangers and opportunities of blockade become clear. In the 1750s, the dockyards of Breast were on the Penfield river, which issues into a large, enclosed harbor. The harbor reaches the sea through a narrow channel, the Goulet, which runs nearly directly east and west through high ground. There are two anchorages outside the Goulet, Berthaume Bay and Camaret Bay, which are both further protected from the Atlantic with reefs, rocks and islands, and there are three passages to the ocean from those anchorages. The Iroise is to the west, and is scattered with rocks; the Four passage to the north leads to the Channel but it is narrow and beset with a very fast tide-race, and to the south is the Raz de Sein, a very narrow passage through a set of reefs with a rock right in the middle of the northern end of the passage.

The tides flow through those passages at varying rates: the Goulet at 3 knots (nautical miles per hour), the Four at 4.5 knots and the Raz du Sein at 7 knots. The distance from the Goulet to the Raz is 25 miles, so unless a fleet had very exact timing it is nearly impossible to make the trip from the ocean into the harbor or vice versa except with exact timing, which means that ships had to anchor in one of the bays (Berthaume or Camaret) to wait for a tidal change.

This both complicated and simplified the task for the British. There was no one point in the sea from which to watch all three passages except for close in to the Goulet, but there was also no high ground at the western end of the Goulet for watchers to see a blockade fleet further offshore. The winds in the region generally blow from the southwest, which means that it was possible for the French to enter the Goulet most of the year, but leaving required an easterly or northerly wind, which meant the French usually used the Raz de Sein more than the other channels both for entering and leaving.

The French also used the Raz because, in the days before latitude was easy to find, ships usually approached a port by finding a landfall at a line of longitude (an east-west parallel) then "running down" that line until they saw a landmark. For the French, the simplest landfall was to Belle Isle (southwest of the Goulet) and then bearing up on the port tack to Brest or the starboard tack to Rochefort or Bordeaux.

An armchair admiral, then, would assume the best place to put a blockading fleet was to the southwest of Brest, near Belle Isle. The problem with that, though, is that any westerly gale would give the British a lee shore to the east which they would have to escape by heading to the southeast, into the Bay of Biscay and away from home. The British fleet in fact had to be kept to the west or northwest of Ushant, so that in case of westerlies they could seek refuge in one of the Channel ports (usually Torbay in Devon). The unfortunate fact of that is that a fleet in that spot can't keep track of the Raz, so the offshore fleet would have to be stationed there with an inshore squadron able to pass messages to the offshore fleet and sound an alarm if the French tried to break out.

This is exactly what British admiral Sir Edward Hawke did in 1759: the bulk of his fleet lay off the northwest of Ushant, with two small ships of the line under Augustus Hervey anchored off the Black Rocks at the Iroise watching the Goulet. His ships were often blown off station, but a westerly wind usually meant that the French were bottled up in port even as the British ships were blown off blockade.

This is something I'm very interested in. I sail and I'm very interested in how things worked during the age of sail. In my limited experience, undocking is relatively easy, some combination of tidal current and wind can ease the ship into the main deeper channels where the ship then gets more options on sail direction.

Docking on the other hand is really tricky, especially a ship that is heavily loaded with cargo. The best way to dock is a situation where the tide and wind are acting in opposite directions. Use the wind to build speed and momentum to get to dock position, then prior to docking strike the sail and let the tide act as a brake to kill the momentum to get into dock position and just touch the dock. The dock piling can't take repeated hard hits from a ship without damage and the ship itself could get damage accumulating over time from hard docking. I would like to see some reviews of old sailing ports and see some illustrations on how it was all done. I'm curious about things like whether the incoming ships collected together somewhere then the outgoing ships all left at the same time and incoming ships at the collection area headed to the dock at the same time.

A crane is one of the simple, or basic, machines. It can be as rudimentary as a winch and a pulley. You can lift enormous weights with a single man's strength if you set it up correctly.