You're actually looking to maximize time in a sector without ever letting the rebel fleet catch up to you. This requires some planning. For the first few jumps, you can ignore the rebel fleet which is far behind you. What you really want to do is chart a course into an area dense with locations so you have a lot of options for places to go before you pop out of the sector. When the rebel fleet is close (say, 4 jumps) try to visualize the progression of the red zone and determine how many jumps you have left, then try to visit as many places as possible before you have to vamoose.
For choices, most blue choices are good. Choices that involve sending your team somewhere are usually risky and not worth it. When you can, you should always choose to engage a ship rather than back off, unless you can't afford the damage or if they're offering a good deal.
Your first hundred scrap should go into either another layer of shields (2 in shields, 2 in reactors to power it up), OR a really good weapon, and then getting that weapon's requirements. As a rule, lasers are better than missiles, which are better than Ion, which are better than beams. Avoid Bombs, Drones and other submodules (crew teleport, hacking, mind control) until you understand more about the game. Repair whenever possible, but you should focus on building up your core (defense and offense), before repairing. The same applies to fuel, you should always try to be at around 10-12 fuel for worst-case scenarios. Avoid spamming missiles, unless you really need to.
Focus fire on enemy weapon systems to knock them out, saving you a lot of scrap in repairs. Try to get 6-8 crew before the final fight. When you can get into a stalemate, do so, so you can level up your guys for free. Always keep someone on piloting, you lose all evasion if no one is at the helm, unless you get levels in piloting. In general, you want a Pilot, a Weapons guy, a Shields guy, an Engines guy, and two anti-boarder guys (Mantis, Rock, Humans will do in a pinch). Pause often, especially in cases of boarding, to determine what are your optimal moves. Use the doors to evacuate air from rooms to choke boarders and fires.
While most unpowered systems (Piloting, Doors, Sensors) can be unleveled, a single level in Doors goes a long way. Focus on getting Shields to 3 before pumping into evasion. Maintain your weapons progression whenever possible, you want to be able to break a 4 layered shield before section 8, Missiles help a lot at shield breaking, but are expensive, so it's best to conserve your missiles, using it maybe once or twice per fight to take out key systems (Shields, Engine, etc). The later in the game you go, the more free missiles you build up, at which point you can play a little faster and looser with them. It's usually best to not use up scrap whenever you get it, but only at a store, for a weapon/crew/fuel/repair, or at the end of a sector. Stockpile as much scrap as you can handle to maximize your options.