Sector priorities in FTL
Advanced Edition
Generally, I'd go with Red > Green > Purple for sectors you can't see. When choosing between two specific options, read on further.
Sometimes, your choice of option may determine a further path. For example, if you're forced down 3 purple sectors, then choosing a Civilian over a Mantis sector to avoid them instead is worth it considering how low down the list the purple sectors are.
The choice of sector depends on your current situation. Are you low on health, then go to a "benign" sector. In almost all other cases, we'd like to go to a 'Rewarding' sector. My personal order for the sectors is:
Danger rating (likelyhood of damage to your ship, from least to most):
01. Engi sectors
02. Civilian sectors
03. Zoltan sectors
04. Nebulas
05. Pirate sectors
06. Rebel sectors
07. Slug sectors
08. Mantis sectors
09. Abandoned sectors
10. Rock sectors
In general, this is a combination of events and the likelyhood of stores appearing. Green sectors can have 3 stores and usually have at least two, while red sectors may have none. Slug sectors have a lot of annoying events so are pretty low down the list. Abandoned sectors just tend to have very powerful enemies in Lanius Ships. Rock sectors tend to be the worst. A lot of enemy ships, many have missile launchers, and a lot of hazards as well, combined with little stores. Pirates are a red sector, but the enemies tend to be pretty weak.
Reward value (from most to least):
01. Crystal Sector, Rock Homeworlds*
02. Zoltan Homeworlds**
03. Pirate sectors
04. Mantis or Rock Homeworlds, Rebel Base
05. Mantis, Rock, and Rebel sectors
06. Engi Homeworlds***
07. Zoltan sectors
08. Civilian sectors
09. Engi Sectors
10. Abandoned sectors
11. Slug Home Nebula
12. Slug Nebula or Nebula
*With a 'Crystal' crew member.
**On a non-zoltan ship.
***Requires an engi crew member
The crystal sector is #1, because it's free. Rock Homeworlds are two sectors in one, so a lot of scrap reward, provided you have a crystal crew member (and didn't forget to (re-)name him "Ruwen").
After that, Zoltan Homeworlds are the best, since they are the only place you can get a Zoltan Shield.
This is followed by Pirate sectors. Pirates tend to give increased rewards (double surrender offers). Slavers are some of the best opponents to fight early on, and you get a lot of them in a Pirate sector.
This is followed by the Red homeworld sectors. Red homeworlds give about as much scrap as a regular Red sector, but also have a nice quest with a double reward, so are slightly ahead of the normal version. You do need to be able to do the quest (have the right system upgrades).
After that is an amalgamation of the red sectors. A lot of ship battles increase their scrap value.
Then comes the Engi Homeworlds. The ship quest often has very good rewards, like a free weapon, and adds two combat beacons. In addition it's easy to do compared to most other ship quests on a regular run.
After this, the green sectors. They're placed in that order since the Zoltan sectors most often have Nebula beacons. Nebula beacons are a good thing in normal sectors, especially with long-ranged scanners. Picking up auto-scout fights at half the fleet advancement cost is great scrap per jump value.
Now the Abandoned Sector. While it is a red sector with quite some ship battles, the very poor quality of events and the absence of nebula beacons which give double jumps of somewhat less quality in non-purple sectors make this a low-rank anomaly.
Finally, at the bottom the 'nebula' sectors. An entirely nebula sector reduces the fleet advancement to 75%, which is not really enough for the nebula beacons to be worth it. There are only so many beacons in a sector, and just not enough fights in the sector in total, even if you do get extra jumps, you often can't use them well in these sectors. So you spend more fuel and don't gain more scrap. Slug Homeworlds is also not that great; the quest is not worth doing since you have to give up on a lot of scrap for not destroying slug ships often getting nothing from their 'storesss'. The only real advantage is more often finding the 'moons' quest that gives you a crew member.
The answer by Aventinus covers the more specific point of making sure you can beat the enemies in the sector. This answer kind of assumes you can, however you may adjust the values accordingly. E.g. going to sector 3 mantis homeworlds with a 1-crew engi-B vessel or going to Lanius sectors with a boarding ship aren't good choices. But provided you can beat the enemies you should go by reward order. Just cut the sectors you think are too hard for you to handle (this comes with experience and depends a lot on your run).
Captain's edition
This changes things up a bit. While the Red > Green > Purple rule still holds, there's a few more exceptions so the rule isn't as 'powerful'.
Danger level (least to most):
01. Engi sectors
02. Civilian sectors
03. Quarantained Sectors
04. Zoltan sectors
05. Nebulas
06. Industrial sectors
07. Slug sectors
08. Auto sectors
09. Mantis sectors
10. Abandoned sectors
11. Pirate sectors
12. Rebel sectors
13. Hazard sectors
14. Rock sectors
41. Federation sectors
42. AI sectors
Do not underestimate the AI and Federation sectors. They are very dangerous. In a sector 7 AI or Federation sector, expect every fight to be as hard as a Flagship stage, or worse. The nano-bot clouds in particular can render your ship quite useless. Rebel, Pirate and Rock sectors have also been souped up with ASB events and potentially very dangerous enemies (rock minelayer wpn refit*, pirate cruiser, Bounty Hunter, Laser Barge*).
*Also known as the guys with 3-4 burst lasers every time
Hazard sectors can also be quite nasty because of the large amount of unavoidable damage they usually contain.
Reward value (most to least):
01. Crystal, Rock Homeworlds*
02. Pirate, Pirate Bases, Federation, Rebel Base, AI***, Auto***
03. Mantis or Rock Homeworlds
04. Industrial, Mantis, Rock, and Rebel
05. Zoltan Homeworlds
06. Civilian Coreworlds
07. Engi Homeworlds**
08. Zoltan
09. Civilian
10. Engi
11. Auto***
12. Abandoned
13. Slug Home Nebula
14. Slug Nebula or Nebula
15. AI***
??. Quarantained***
*With a 'Crystal' crew member.
**Requires an engi crew member
***Better with some specific preparation, see below
Reward value also fluctuates based on the "Trading" system. Basically it's best to gamble for things that are popular in sectors near the top of the list, for a double reward (better sector, and expensive selling).
For the new sectors, the Auto and AI sectors are rated pretty low. While there's plenty of enemies with good rewards, you usually can't get those rewards because of their rediculously fast FTL charge when they run away. And run away they do very often. In addition their augments tend to help them at running away. A lot.
The factory sector is pretty nice. The acid fields have increased rewards and you get more ship fights than regular nebulas. You do need a bit of hull for it.
The 'Quarantained' sector is a real doozy most of the time, with almost no scrap rewards in it whatsoever. Most beacons are either crew lotteries or empty. If you can afford an upgraded medbay and can take some medical supplies and firearms into them you can get some guaranteed crew and scrap at little risk. In general unless you have these specific things it's a very bad sector.
Zoltan homeworlds have moved down a few spots as the Quest no longer gives you a free zoltan shield, but just a random weapon.
Finally, the pirate base and fed sector are regular red sectors, just with a bit more danger level to it.
Rebel cruisers and Pirate cruisers also increase the value of the Rebel and Pirate sectors. Manned rebel cruisers tend not to run away. (The auto variants are cowards though).
If you manage to get an OP ship (e.g. weapon preigniter, glaive beam, flak II), that can one-shot enemies or do 12+ damage within 10 seconds (you can have some wind-up time), you can move up the AI sectors and auto sector to share the spot with the regular red sectors. If you can kill the ships, auto sectors are quite nice as acid nebulas offer increased rewards at the cost of a very small amount of hull (2 points per jump usually).
The Homeworlds sectors appear once per game. Each of them has a special quest for unlocking a new ship.
The Nebulas are slightly harder, especially if you have a weak Reactor, but they can give you higher rewards.
In each sector controlled by a specific race, you will encounter more ships of that race. Mantis like to board you, Rocks tend to have missiles and beams, etc. So choose your sector according to what type of opponent you are strong against.
You can find more on the wiki.
As already mentioned you want to visit the Homeworlds since they offer special quests.
Other than that there's no universal way of moving through sectors but you want to bear in mind two things:
You should keep in mind the weaknesses of your ship. For instance, let's say that you use the Rock Cruiser A which starts with only missile guns. At this point you want to avoid a Zoltan sector because Zoltans use Zoltan shields which are impenetrable.So you choose your path depending on your ship and by knowing the strategies of various races.
The second thing to bear in mind is the pros of visiting a sector. For instance, let's say that you have invested in a ship which uses Drones. If you want to upgrade your equipment, visiting an Engi sector will offer more high quality options in the sector's markets. Or, for example, if you have a great amount of fuel you can invest that by visiting a nebula sector. When you're in nebulae the progress of the rebel army is halved. This is way you can visit many stars and increase your earnings.