XCOM: Enemy Unknown

*** Please note, the numbers shown apply to a "Normal" difficulty campaign. For harder levels, the amounts are (approximately) halved.

The fastest way to acquire engineers is to launch satellites over countries that will provide you with engineers every month for maintaining those satellites.

The continent that is engineer-heavy is "Asia" (I use the quotes as, for purposes of this game, Australia is included in Asia), as you will receive an extra 8 engineers per month for having satellites over all four Asian council nations.

You will also receive engineers every month for maintaining satellites over other countries, in the following scheme:

+3 engineers/month for 3 satellites over the African nations.
+4 engineers/month for 3 satellites over the North American nations.
+2 engineers/month for 2 satellites over the South American nations.

Europe offers no engineers, as they are scientist-heavy in their reward structure, which you will need for

As you can see, once you have a full network of satellites up, it won't take many months for you to reach 80 engineers (at +17 engineers per month, in addition to whatever bonus engineers you choose to play for).

One may also acquire engineers by building Workshops, which add 5 engineers (regardless of difficulty level), or by receiving rewards for certain types of missions and council requests. All such rewards are one-time rewards, however, whereas the satellite coverage bonuses repeat every month.

There are four ways to get engineers:
1) Via the Council's evaluation on the 1st of each month
2) Build workshops
3) Fulfill Council requests that offer engineers as a reward
4) Perform missions that offer engineers as a reward

Monthly Council evaluation - On the 1st of each month, the Council will give you an allowance of money, scientists, and engineers based on which continents you have satellites over, and how many, and which difficulty level you're playing on. In every case, Asia gives you the most engineers per month, followed by North America, South America, and Africa, in that order. Europe doesn't offer engineers. So, obviously you want to build your base in Asia and focus placing your satellites there. You can read the map.

Workshops - Each workshop you build adds 5 engineers to your staff. You start the game with 1 workshop and 5 engineers. To get your next workshop, you must have 6 engineers on staff. That means you have to pick up at least one engineer from one of the other three ways before you can build any workshops. You need 15 engineers to build your third workshop, then after that the requirement goes up by 10 for each suceeding workshop, i.e. 25 for the fourth workshop, 35 for the fifth, etc.

Council Requests - At random times, random nations will ask random things of you in trade and offer you random rewards. For example, Mexico might ask you for 8 laser pistols for $200, or Nigeria might ask you for 8 floater corpses for 3 scientists. Sometimes the reward is engineers. Your best bet with these is to use the gray market sparingly, so that you'll have enough of whatever they need on hand. You have 20 days to fulfill a Council Request before it expires. Obviously, if you're going for the 80 engineers achievement, you'll want to try to fulfill most of the requests where engineers are offered. When they offer cash, you'll probably want to take those as well, because cash is usually tight. When a country offers you scientists, you can either take them, or tell them to suck rocks. Your call.

Missions - Some missions offer engineers as a reward. The usual place you will see this is Alien Abduction missions, where you have to choose which country to assist. The reward is always either cash, engineers, scientists, or soldiers, and you usually have to choose from three of those, so you have a roughly 3/4 chance every time there's an alien abduction of getting a chance to pick up engineers (usually (always?) 5 of them). For political reasons, you won't always be able to choose the country that gives you the reward you want, or you may really, really need cash or a new Assault Captain or whatever some other nation is offering. I usually base my decision on which country to help on panic levels, rather than rewards, and only go by the reward when it's a political toss-up. There are also a few missions, like VIP escorts, where there is only one choice and only one reward.

Here are some specific steps to get you going in the right direction:
1. Put your base in Asia.
2. Construct a satellite and launch it over Asia.
3. Make your first three facilities a Power Generator, Officer Training School, and Workshop. Don't build in the spot next to your Satellite Uplink; leave it empty. Don't move on to Alien Containment until all three of those are completed. Hopefully you can get engineers from a mission reward during March, but worst case, you'll be able to start on that workshop on April 1, after your first Council evaluation.
4. Once you have 10 engineers, build a Satellite Uplink next to your existing one. That gives you the capacity for 3 more satellites.
5. Launch at least one of your next 3 satellites over Asia. I'm saying "at least one" because realistically, you're going to have to pay attention to some other nations before too many of them leave the Council.

You can take it from there. In my Impossible playthrough, I got to 80 engineers and the achievement with my December Council evaluation. I had 5 engineers coming in per month from satellites, and I had 5 workshops. On easier difficulties, everything should be about the same only you'll get the achievement sooner, because the satellites produce more engineers and you have more leeway in choosing Alien Abduction missions (because nations don't get hysterical so quickly at lower difficulties).