I wrote most of this up as an off-topic response in another post. And it seemed important enough for new players to warrant its own post. I do not have Odyssey so cannot provide any information on farming Odyssey mats.
1. Why engineer?
Engineering lets you make upgrades to your ship's modules to improve them, in many cases doubling their performance. A full description is beyond the scope of this guide. I suggest Fox's guide for unlocking the engineers for a procedural guide. And Inara's engineer page as a quick reference for the available engineering upgrades, which modules each engineer can upgrade, as well as the engineering mat cost. Note that each column of engineers on that page represents an unlock chain. The engineers above unlock the engineers below. The down arrow between portraits means you must unlock the above engineer to access the ones below. The square between portraits means that engineer is at the same unlock tier as the one above (unlocked by the engineer above the down arrow).
Warning: The way engineering works in the game was completely revamped several years ago (2017 I think?). So a Google or YouTube search may turn up old guides and videos which are no longer relevant. Be sure any reference you find dates from within the last ~3 years.
2. What are engineering mats?
Engineering materials (mats) are the currency you use to pay for engineering upgrades on a module. They're also used to pay for modules from technology brokers. These are mostly modules from human technology brokers which have been added to the game. Notably, a double-engineered 5A FSD (better jump range than if you fully engineer a 5A FSD yourself), and a detailed surface scanner with 2x the probe radius. But a few of the Guardian modules require a few engineering mats.
You can get mats from shooting geological and biological features on planets and moons, shooting cargo racks at crash sites, gathering debris lying around settlements, from mining, scanning radios and beacons, in combat debris, and as mission rewards. There are three types – raw, manufactured, and encoded – and several families within each type of varying grades (G1-G5). They have their own separate section in your inventory (right side menu), and do not take up any weight or space. Although there is a maximum limit to how much of each mat you can have (300 of G1, 250 of G2, 200 of G3, 150 of G4, and 100 of G5). Unfortunately the wiki page on materials is rather poor. The best reference for all the materials is actually the wiki page on material traders.
Each engineering upgrade costs different mats, which you have to pay each time you perform the upgrade (for a random amount of improvement). It is a PITA keeping track of all of them. People seem to use two ways to figure out how many mats to gather.
Build out a ship on coriolis.io or edsy.org. After you add the engineering mods you want, both sites will estimate the type and number of engineering mats you will need to create that ship. Then go and collect those mats.
In coriolis.io, click the gear icon in the upper right.
In edsy.org, click Analysis at the top, then click on Generate Retrofit Report.
Tools like EDEngineer can help you keep track. ED Discovery has a page which shows you your current engineering mat inventory at a glance (no need to dig through the right hand menu).
Just max out on top-tier G5 mats, and use material traders to convert a bunch of those to lower grade mats so you have a bunch of everything. When you need more lower grade mats, visit a material trader and trade in more G5 mats. When you run low on G5 mats, go farm some more.
3. How do I use material traders?
A material trader is one of the contacts at certain starports who will let you trade one engineering mat for another. Each one only deals with one type of mat (raw, manufactured, or encoded). Use Inara's nearest station search to find the material traders closest to your position. eddb.io can search for them too, but does not tell you the type. So you must use Inara.
The trade ratio is most favorable when trading within the same family (a row in the material traders wiki page), and when trading down from high-grade to low-grade (towards the left on the material traders wiki page).
When you trade down in grade within a family, you trade at a 1:3 ratio for each grade. e.g. Trading 1 G5 mat gets you 3 G4 mats, 9 G3 mats, 27 G2 mats, or 81 G1 mats. Due to this steep ratio, it's generally not worth farming G1, G2, or G3 mats. Most people farm G5 and just trade those down.
Trading up within a family happens at a 6:1 ratio (if you trade down and back up, you lose half the mats). e.g. You need to trade 6 G1 mats to get 1 G2 mat. 36 G1 mats to get 1 G3 mat, 216 G1 mats to get 1 G4 mats, and (if you had the capacity) 1296 G1 mats to get 1 G5 mat. Don't trade up unless you absolutely have to.
Trading from one family to another also happens at a 6:1 ratio at the same grade. e.g. You need 6 biotech conductors (G5) to get 1 pharmaceutical isolators (G5). Or 6 conductive components (G2) to get 1 chemical processor (G2). Don't trade between families unless you absolutely have to, although sometimes it will be a better option than farming the other mat. It's also the only way to get rid of excess lower tier mats if you don't need any more mats within that family.
Go to Jameson Crash Site at HIP 12099 1 B. Scan the four beacons there. Most people do it in an SRV, but it can be done from your ship if you're OK with 2-axis rotations. Then logout and login again and repeat (this is the part everyone hates). That gets you encoded mats in the adaptive encryptors capture family, most of them G5. Trade those to other G5 families. The 6:1 ratio sucks, but it's usually the fastest way.
Use a frame shift wake scanner, and sit outside a starport scanning wakes. I haven't had much luck with this, but some people swear by it. You can engineer the wake scanner as long range to reduce the amount of moving around you need to do. (This increases the power needed by a lot, so make sure you have a big enough power plant.)
You can also do missions which give modified embedded firmware (G5) as a reward, then trade those as needed. The higher rep you have with a faction, the more of these missions appear, and the more G5 mats you get as a reward.
Max is G4, not G5. Though some are incorrectly labeled in the game as G5.
Farm the crystalline shard spires at HIP 36601 and Outotz LS-K D8-3. Note that the moons are about 150k ls and 300k ls from the primary star, so be prepared to spend some time in supercruise. Fortunately once you get out that far, all the moons in that system are near each other. Land at one of the biological sites listed in the quick reference guide (they're the flattest). And shoot the shards off the crystalline spires to gather G4 raw mats.
Note that there's a bug which sometimes limits you to collecting only 100 G4 raw mats. If this happens, just logout and back in again, and your inventory limit should correctly reset to 150.
These systems are about 1500 ly from the Bubble. Fleet carriers regularly make the trip out there. You can join the Fleet Carrier Owner's Club discord for trip announcements and schedules. (For some reason everyone takes their carrier to HIP 36601 first, when it makes more sense to go to Outotz LS-K D8-3 first. That way anyone wanting to jump to the other system ahead of schedule, or wishing to meet up with the carrier for the trip back to the Bubble, only needs to fly 150k ls in supercruise. If the carrier goes to HIP 36601 first, they need to fly 300k ls.)
There are no selenium crystalline shard spires. Selenium was originally coded as a lower tier mat (G2 I think), and spires can only give G4 mats. Shortly after the game was released, raw materials were overhauled (why some of them incorrectly show up as G5). Selenium was upgraded to G4. But the game still thinks they're G2 – from what I understand, fixing that would alter the procedural generation, and change the entire galaxy.
Instead, use edtools' selenium finder. Use it to find a nearby system with high selenium share (percent of common raw mats which are selenium). Go to that world, scan it with the DSS, find a geological site which is relatively flat and land there. Scurry around in your SRV shooting up crystalline fragments – these will yield selenium in proportion to the selenium share (typically 50%-70%). When you can't find more fragments, lift off and go to a different site. I timed it and it's about 2-3x faster than farming crystalline shards and trading for selenium. It's also a lot more fun since you can drive the SRV at full speed. And because the sites are in the Bubble, you can do it on a whim and only need to gather as much selenium as you need.
Note that needle crystals frequently give other G4 mats, so shoot any of those you see too.
Farm high grade emissions (HGEs). Equip collector limpets and a cargo rack. Fly into a system, scan the nav beacon, and check the signal sources in the nav panel for HGEs. Fly out to one and there will be a few G5 mats. The type of mat depends on the system and faction state (check the quick reference). Now the part everyone hates: Completely exit the game. Logging out is not enough, you need to completely close the game. Then restart it. Login, jump to supercruise. Check around, and the HGE will be nearby. Drop back into it and there will be more G5 mats.
You can also do missions which give biotech conductors and exquisite focus crystals (G5) as a reward, and trade those for other mats. Pharmaceutical isolators and military supercapacitors are the rare G5s from HGEs. Proto radiolic alloys are also a pain because they share a drop table with other G5 mats, and you always seem to get the wrong ones. You will probably want to trade for these G5s rather than farm HGEs. The higher rep you have with a faction, the more of these missions appear, and the more G5 mats you get as a reward.
In general, it's not worth stopping to collect the mats you get from combat kills – they're mostly lower grade mats which take too long to collect. If you specifically target a G5 mat, the limpet suicides after collecting just the one mat. You'll get mats more quickly by stacking a bunch of RES massacre bounty missions which give biotech conductors and exquisite focus crystals as a reward. Completing those missions as quickly as possible (i.e. not stopping to collect mats from kills) usually gives you manufactured mats quicker, even if you have to trade at a 6:1 ratio.
Note that these are the "optimal" ways to farm engineering mats. There is nothing wrong with doing different methods of farming if you don't care about being optimal.
I did Dav's Hope a lot (Hyades Sector DR-V c2-23 A 5) when I first started playing. It was tremendously helpful in helping me learn to drive the SRV, as you can time yourself as you make each circuit. The story at Dav's Hope is heartbreaking too.
I highly recommend everyone visit the Bug Killer crashed Anaconda site (HIP 16613 1 A) at least once. Preferably at night for the creepy ambiance and story ;).