Access

Once invited, the engineer is going to demand something from the CMDR. In-game there's no way to learn this except by visiting the engineer's station, landing, seeing the demand, and then flying away again to fulfill it. Don't do that. Look up what the demand is first. Farseer above demands 1 unit of Meta Alloys; Inara will show you that you can buy those at the Darnelle's Progress station in Maia. Many engineers want rare goods that are similarly single-sourced, and some want more of those goods than the station will sell at once. Make multiple trips or contact the East India Company.

Ranking Up

At last the engineer is going to actually do their job! But at first, all they'll offer are Grade 1 modifications. Each engineer has to have their higher grades unlocked; this unlock is for every modification the engineer offers. Raise Tod McQuinn to Grade 5 with multi-cannons and he'll offer Grade 5 for railguns, too. Engineers have a variety of ways to unlock higher grades -- turning in bounties, selling cargo, etc -- but only two are worthwhile. Selling exploration data and performing modifications.

Selling exploration data is straightforward: turn in $24M of exploration data at the engineer (after unlocking access) and the engineer will progress straight to Grade 5. Technically trade profit, bounties, and combat bonds all work the same way, but it's much faster to get $24M of exploration data than that number of any of the others. Generally where possible, and particularly starting out, use exploration data wherever possible to unlock engineers. These engineers include Farseer, Martuuk, Jameson, Jean, Tah, Sarge, and Sedesi.

Performing modifications is predictable but does require access to engineering materials. Wherever possible, perform modifications you actually want on your ship; where not, perform modifications based on what materials you can afford to waste. You need to perform three modifications at the current accessible grade to unlock the next grade -- that is, performing three Grade 1 mods unlocks Grade 2, and performing three Grade 4 mods unlocks Grade 5. Lower-level mods can unlock much higher Grades, but the experience required scales exponentially. It could take dozens or hundreds of Grade 1 mods to unlock Grade 5. The data-based scaling is $160k to unlock G2, $740k for G3, $3.2M for G4, and $20M (for $24M total) for G5. That's 400 - 500 G1 mods if the scaling is consistent.

That said, more than three mods per Grade are required as a practical matter.

Module Pacing

In order to apply a Grade 2 or higher modification to a module, that module must have that same modification completed at the prior level to at least 80%. Every module modified to Grade 5 must first progress through Grades 1, 2, 3, and 4. While an engineer is not fully unlocked to Grade 5, their modifications give less progress towards 100% of the possible bonus on a per-modification basis. Thus, to be assured of having enough materials to fully unlock an engineer, a CMDR should visit with materials to perform six modifications at each intermediate grade.

That is, to unlock an engineer but not perform a Grade 5 modification, plan on materials to perform six Grade 1 mods, six Grade 2 mods, six Grade 3 mods, and three Grade 4 mods. In my experience unlocking the engineers, I've never needed more than five mods at any level, and rarely more than four, so six should be iron-clad.

To move straight into a Grade 5 mod, bring materials for six Grade 4 mods. In all cases, remember that there's no reason to progress a modification at a given grade beyond 80% (at which point the option for the next higher grade will become available) if the plan is to modify at the higher grade.

Note also: if the goal is to modify multiple modules on first visit (suppose the plan is to fully engineer three shield boosters at Vatermann), fully unlock the engineer with the first module before doing and modifications on the others. The engineer's low-grade modifications will cover more of the possible benefit the higher the engineer's grade, so one at a time is more material-efficient. This effect makes it worth ranking all engineers up to Grade 5, even those like Qwent and Dekker that don't have Grade 5 modifications available.