Are you playing Easy or Normal? The advice won't change much, but be aware that "easy" difficulty is actually quite tough and normal is very difficult.
Your first problem lies in "avoiding conflict". You must maximize the amount of scrap you take in. It's worth pushing it to get every last piece. On easy there's a certain buffer you have that you won't on normal, but even there you are looking to collect scrap at every opportunity. Fight everything you can. Fight when you get a chance to run unless you are near dead. If someone offers to surrender, you'll have to do your own calculations. Surrendering generally gives more consumeables (fuel, missiles, drone parts) than scrap; destroying an enemy gives more scrap and less of the other stuff. You need to think in terms of maximizing your scrap intake. Hit as many planets in a sector as possible and fight as much as you can.
You must also walk a fine line. You do want to horde scrap, but you need to spend some scrap to help in the early going. Generally, you spend early scrap where it's most cost-effective.
Engine Upgrades: generally speaking, you want to upgrade your engines quickly. Engine upgrades snowball; each engine upgrade increases dodge by 5% (up to a breakpoint at like 6 levels of engines). Your pilot and engines crew can skill up whenever your ship dodges. Once they gain a level, your dodge chance goes up, and so on and so forth. Engine upgrades are cheap. At the start of the game 1 level of engine upgrades (and the reactor upgrade to power it) is easily less than half the cost to add 1 point of shielding. The first couple of engine upgrades are some of the best scrap you will spend. I get 1 very quickly, and usually have 3-4 by the end of sector 2 at the latest. Dodge is the only defense against ballistic (missiles, bombs) weaponry, and is also useful against regular energy weapons (which are the most common type of weapon). Useless against beams, though.
Weapons: if you can find a weapon, upgrading your offensive capabilities is an excellent idea. I like to hold scrap in reserve for this possibility. The next store might have a Blaster Mk I or II, and these are typically must-buy for me. Always be on the lookout for more weapons. Weapon upgrades get expensive fast. Most of your scrap saving is about hoping for weapon upgrades, or saving for things like drone control/crew teleporter. As to what to look for, Blasters Mk I and II are exceptional items (being very efficient from a power/system standpoint, requiring 2 weapons/power each). Small Bombs, Ion Bombs, and Artemis Missiles all require 1/1/ (system upgrade/power) but do at least 2 damage (Ion Bombs do 4 ion damage).
Shields: You can ignore these at first. Shield upgrades are great- they're the only beam defense, remember, and they're really helpful in asteroid fields. However it's a pricey upgrade. I won't touch this until I have at least 4 points of engines, typically. And I usually don't get a third point of shields until late (that upgrade is likely to cost 150-160+ scrap in total).
Systems: Both the crew teleporter and the drone control give you a lot of flexibility. You'll need to find drones and crew-entities to use with them, of course (drone control always comes with one drone though). These upgrades cannot help you out that much if purchased early. Note, they are very common, so you don't need to worry about passing on them early. By the end of sector 2 I'm saving scrap to get one or the other, if not both. Note that the teleporter, once you can use two decent crew in tandem, will really maximize scrap gains. Capturing ships gives the best rewards.
Augments: I consider automated re-loader a must buy (pre-igniter as well, but it's super expensive). However, augments generally only help in the long term. Still, I will alter early game plans to buy a choice one (I'll get a scrap arm as well; if you plan on going drone heavy a drone arm is nice. I usually avoid the repair arm).
Crew: adding crew can be helpful. If you start with under 4, you should get to 4 asap to work the "primary" systems (pilot, weapons, shields, engines). If you want to use teleporters, keep an eye out for more. Mantis and Rocks are the best "combat" crew (Mantis do extra damage, Rocks have 150 hp). Note that when you fight slavers, they will typically offer you a slave to let them live. This can be an excellent way to pick up crew early. Consider the cost benefits; the cheapest crew is a human at 40 scrap, iirc. The most expensive cost 65 scrap, and it's not common to find them at stores. In the first sector, killing a ship probably nets you under 40 scrap. So if a slaver offers, take the deal (even if it feels a little icky)!
Sub systems: do not neglect your doors! If you start with doors 1, the first upgrade is 20 or 25 scrap. I don't do it on sector 1 but I usually get it in sector 2. This is a huge help in controlling fires and dealing with boarders. I don't always get the third point, and when I do it's usually late. Sensors 2 can be useful in some circumstances. Doors upgrades are great though.