All of the small blue pieces are in play. Players usually trade their small ships to get new technologies, but Alpha will have to trade their large ship to get blue.
You should always start with green (either as a homestar or starting ship). Otherwise, you'll never be able to build a ship and you can't win. Sure you could start with red, yellow, and blue and then trade your ship to a green, but you may as well have started with a green ship.
You should almost certainly start with blue. Blue at home makes it easier to get ships in a variety of colors, but you could technically discover a blue system after a couple of moves and get your color variety from there.
Most players pick a large green as their first ship (hoping to sacrifice it at some point to beat their opponent at getting more large ships). That means they want one homestar to be blue, so the first color question they face is: should the other star be yellow (what Andy calls the "quick start") or red ("planetary defense system")?
I don't think that getting to move a ship out of your home one turn earlier is the real reason to pick a yellow star. Instead, having a yellow star makes it more difficult to destroy your home by star demolition. Demolishing a yellow star with a Doomsday machine requires a big yellow-economy advantage. Consider what it takes to make a 2b/2y Doomsday machine against a yellow-blue home:
One yellow piece is the target homestar, so the 2b/2y Doomsday machine must have 5 of the other 8 yellow pieces as ships. If the defender can get 4 yellow pieces as ships or stars, the 5 yellows needed for the Doomsday machine won't be available. Of course, the attacker might try to rebuild their sacrificed y2 after the blue stage...
One advantage of having a red star in your home is easy to understand: you can capture any lone invader as long as you have a large ship. Another advantage is that your red star is almost impervious to three-turn Doomsday machines (see this section). But to me, the biggest advantage of a red homestar is that it is not yellow. By not having a yellow star in your home, it is easier to produce yellow ships in your home, and I consider yellow the most important economy to keep strong.
The homeworld size options are named here.
The banker setup is named for the investments that you can make by simply moving a green ship out of your home. In my games, investment systems usually get invaded before they can be cashed in, but it does seem like bankers successfully invest the most often.
As you read the game rules for the first time, you may think that the Goldilocks home setup offers some of the advantages of the banker and fortress. I haven't found this to be true. Except for a strange late-game situation, I've never seen anyone deliberately invest in a medium star, so I'd say Goldilocks does not provide better investment opportunities than the fortress. Although the supply of medium pieces may run out, I don't recall any games where a Goldilocks system was difficult to reach because of a low medium-supply. I think of the name "Goldilocks" referring to the simple fact that the connected stars are medium: not big and not small (like Baby Bear's chair).
To me, the important aspect of Goldilocks systems comes toward the end of the early game when medium ships are starting to come into play. You can accelerate the economy of your choosing by discovering medium stars of that color and getting access to large ships more quickly.
The strategy guides published by Looney Labs say that the fortress is hard to invade. I have played a few long games where the lack of small stars played an important role, but from what I've seen, the fortress has a more important role on the early game. Fortresses are the only homeworlds that do not have the "small star problem" described later. Fortresses also make it easy to jump-start an economy of your choosing by discovering small stars of that color.
Gemini homeworlds can make the universe small or even connect the homeworlds. For this reason, games with a Gemini homeworld tend to be somewhat shorter than large-universe games. The player who moves first gets a head start on building, and short games give the second player less time to catch up. Theoretically, short games should tend to be better for the first player (I don't have data to back this up, but it's what makes sense). Since Gemini homeworlds can shorten the game, they seem to be good for the first player and bad for the second. As I'm writing in early 2021, the player Draw5PlayAll on SDG has had very good success defending the top spot on the SDG ladder by starting with a Gemini system (they have been lucky enough to have the first move four games in a row!).
Any time that you include a small star in your homeworld, the economy of that star's color gets a little head start: you've removed a small piece of that color, so medium ships of that color may be built sooner. Because of the danger of overpopulation, it is a little more difficult to participate in the economies corresponding to your homestar colors. Thus, small stars give a little boost to an economy where you have a disadvantage (possibly leading to a ship size advantage for your opponent).
In particular, if you are going second and your opponent chose a small star, you probably shouldn't have a small star of the same color (unless you're attempting an Instafreeze). They can get the last small of that color as a ship and freeze you out.
Getting frozen out of a color can quickly lead to a frustratingly hopeless position. On the other side, getting a monopoly can ensure a quick victory. Avoiding freeze-outs is a very important skill.
Here's a high-level description of a typical freeze-out/monpoly. I'll call the player who gets the monopoly M, the player who gets frozen out F, and the color of the monopoly x.
The freeze-out process can happen alarmingly quickly. This is the main reason that I warn people that Homeworlds is a very unforgiving game. Sometimes, F only has a one turn "window" to react to M's monopoly attempt.
This is a trap you can set for new players when you move second.
If your opponent chooses a small star that is blue or green, (or maybe even yellow) you also choose a small star and first ship of that color. You can build the last small piece of that color as a ship and freeze them out of that color (see below).
New players won't notice the problem and you can build mediums and larges of that color before they know what hit them.
Experienced players will respond by trading their initial (large) ship to enter the economy and the position will be about even.
The early game is about getting more and bigger ships than your opponent. Secondarily, you want to get ships in a variety of colors. In particular, it is desirable to have access to every technology in every system you occupy. If you move or trade a ship that was the only piece of its color in your system, then you lost progress in the effort to get every color to every system. Regaining access to that color in that system will cost you a future turn. It is usually better to spend a turn building a ship of that color first and then moving or trading it away.
Even if I have access to a technology through a star, I prefer to have a ship of that color there (as long as there is no risk of my opponent overpopulating). If I manage to gain a near-monopoly of that color, that system may be the perfect place to build an additional ship of that color, even though having two ships and a star of the same color is usually very dangerous.
Another way to waste time is to trade too often. Ideally, will trade for each color once and then build all of the ships of that color you need. Trading later in the game is an important way to manipulate the bank, get large ships of important colors, and quickly open building opportunities in a suddenly-valuable color. However, these things aren't usually part of the early game.
On your first turn, you make your homeworld.
On your second turn, you build a small ship (there's almost never another reasonable choice).
On your third turn, you should consider trading your small ship for another color. If there are any small homestars in the game, then the medium ships of that color will be available for building soon. New players often simply build a third ship of their original ship color on their third turn, even when their opponent is poised to seize a monopoly (often over the blue ships). Don't get left behind! Identify the color(s) that will be open to building mediums soonest and get involved in that economy early.
"Mirroring" is doing approximately the same thing as your opponent in the early game: building the colors they build, trading the colors they trade, expanding when they expand. It may seem unimaginative or like the game is broken, but mirroring is often the best choice for the second player since it ensures that they won't fall too far behind in any economy. In some games, mirroring may be the right choice for a dozen turns or so, especially if the players chose the same colors for their homeworld stars.
Usually, mirroring becomes impossible after a while because the supply of pieces (especially larges) gets unevenly distributed between the players. If you are mirroring your opponent, try to anticipate when they might make a move that you won't be able to mirror. For example, you probably shouldn't build a piece that opens the door to a profitable shopping spree for your opponent. There probably won't usually puts a quick end to mirroring.
I intend to expand this into a small section. See Investment in the glossary