Introduction to Engineers
Each Engineer specialize in upgrading specific modules only. In order to use an Engineer, you must first unlock him/her. At first you get an invitation from the Engineer telling you where to find him or her and how to unlock access.
For example, to gain access to Felicity Farseer, who can upgrade Frame Shift Drives, Engines and Thrusters, you need to have an Exploration rank of Scout or higher. After that you can unlock her services by providing one unit of Meta-Alloys.
After you gain access to an Engineer, you can then start improving your modules using the blueprints provided by the Engineer. At first you will only have access to level 1 blueprints. After making tree of those, the Engineer is upgraded to level 2 blueprints and so on. When you start making level 3 blueprints you will typically get an invitation to another engineer. Blueprints can currently be upgraded to level 5. The higher level blueprints will result in better updates but will also cost more materials including some rare materials that may take some time to get.
Notice: Currently there is a lot of luck involved in getting the best updates; when patch 3.0 is released it will be much easier to upgrade your modules, so I recommend saving your materials until then.
Unlocking Bill Turner
Bill Turner is located in the Alioth system, which require a permit to enter. For this reason it take a little extra time and work to unlock him!
What you need to do is find a system close to Alioth that is controlled by the Alioth Independence faction. By handing in exploration data, bounties or missions for this faction, you will increase your rank with Alioth Independence (and also with the Alliance).
Tip: Go to the Holiacan system - all stations there are controlled by Alioth Independence. There is a compromised nav beacon where you can farm bounties; turning them in is a nice and fast way to improve your rank. Or you can take some of the missions from the stations if you prefer.
When you reach Ally with Alioth Independence check the mission board for a special mission that simply grants you a permit to the Alioth system. If the mission does not show up, take a mission to another station in a nearby system and chances are the Aloth permit mission will show up there.
Next you need to deliver 50t of Bromellite to Bill Turner. A nice place to mine Bromellite close to Alioth is the LHS 2522 system. The rings of LHS 2522 6 B ring is icy rings with pristine reserves, just what we need. If you haven't discovered this system yet, take a trip in your exploration ship first - remember to bring a Surface Scanner. Next load up a lot of limpets and bring your mining ship - I used a Type-7 for mining my Bromellite - check out this Type-7 configuration.
Unlocking Zacariah Nemo
To unlock Zacariah Nemo, you need to be friendly (or ally?) with "Party of Yoru" from the Yoru system. Some people report getting the invite after being ally, others only after becoming ally. The invite mission actually is a little buggy; I was ally but didn't see the mission even after relogging several time.
Workaround: Exit the game - also the game client. Restart and make sure you use a Small or Medium ship, then land at one of the outposts in Yoru (it seems the invite mission doesn't spawn from any of the planetary bases where you can land with a Large ship).
You should then see a mission titled "Engineer Invitation Contract" which is likely a simple data delivery mission to one of the nearby systems.
Unlocking Professor Palin
You need to unlock Marco Qwent first and do some engineering with him to learn about Professor Palin.
Next take your exploration ship out for a trip at least 5.000 ly from your starting point - remember to bring an Advanced Discovery Scanner so you can also get some points towards Elite explorer now you are out there.
Finally you need to bring him 25 units of Sensor Fragments. Go to HIP 17862 and fly to the planet 6 C A and land at the coordinates 30.32 , -98.58 - there is a crashed Thargoid ship and you will find a few Sensor Fragments lying around it. There will also be some alien artefacts lying near the crashed Thargoid Ship - pick them up, drive them away to some open area and shoot them; this will result in 1-2 more Sensor Fragments. (the reason I don't shoot them "inside" the crashed ship is there is a high risk they will roll in under some rock where you can't pick them up). You can switch between Open/Solo/Private Group to spawn more fragments and artefacts.