Making a solution for what to do for the three Elite ranks is challenging, because methods that work, or the best method available, is always changing, but I'm going to keep the following Pastebin updated with what I think are likely the best methods for purely getting the Elite ranks in Combat, Exploration, and Trade. I'm using a Pastebin because the amount of text I'd be putting here would be inane, and the formatting is much better there. Hope this helps.
https://pastebin.com/Ap8GQb5kDULY NOTE: I will be writing and re-writing large portions of this guide to account for changes in 3.1; including, but not limited to: the new Crime & Punishment system, the engineering changes, adjustment of the mission supercruise distance reward modifier, and such like that. Also want to re-write some portions for the sake of clarity. May take a week or two after release of the update.
FURTHER 2022 EDIT: With news that Elite: Dangerous won't be getting further console updates, personal interest has dropped dramatically. So a lot of solutions and such aren't getting rewritten. There's still some useful info on these, but a lot of it is outdated. Sorry about that. Blame Frontier.