Research is expensive and will be painful to acquire before reaching at least ~1 Tvg (1E72) gems. Until then, play as usual. Try to keep excavations at a reasonable price, you will need FCs later to level up the Research Facilities.
When ready, play one of the 6 vanilla factions. It's advised that you select a faction that can easily gain FCs, such as Elf, but most of all: Do not buy the Dwarf or Drow treaties.
Follow the appearing quest tip to unlock the Faction's unique buildings. These quests hint at the requirements, which are a defined amount of a specific building - See table of requirements for exact details.
Once you have bought the Unique buildings and all the faction's upgrades, the upgrade to unlock the faction's related facility will appear. (Fairies are related to the Spellcraft facility, Elfs to Craftsmanship. etc. See Research Facility table.) Unlocking a facility will cost a lot of faction coins (10 Notg (1E121) coins, and 1 M (1E6) Faction coins of the Faction you are playing) and may take time. Don't abuse excavations as you will also need them later to level up the facilities.
Access to research: Once you have at least one facility the "Research" button will appear, giving access to the Research Panel. In future games, the research button will only appear when unique buildings have been purchased.
Leveling up Facilities: Once the facility is unlocked and if you have enough FCs, you can now level it up. Open the research panel and click on the available Facility (big icon) to give it its first RP and make the first upgrade visible. Visible, but not always unlockable by the faction you are playing.
Unlocking each research upgrade: Now, you must find out how to unlock each research upgrade, and by which Faction. In the beginning, several can be unlocked with one run, but eventually, you will need to do a special runs aiming for each specific requirement. (See why_amihere's research guide for tips and spoilers.)
Locked Upgrade (question mark): You cannot unlock this upgrade with the Faction you are playing.
Locked Upgrade (exclamation mark): You can unlock this upgrade with the Faction you are playing. The Icon tool-tip will show your unlock progress %.
Unlocked Upgrade (cross): You cannot use this upgrade with the faction you are playing. (There is an option to hide these icons.)
Unlocked Upgrade (no symbol): You can buy this upgrade but you need more coins.
Do not bother about gem progress for now. Concentrate on adding RPs and unlocking upgrades. (You can buy the upgrades as you unlock them. Most of the time at this stage, you will have no other choice but to use the few available.)
Upon Abdication and Reincarnation, RPs will persist, and the Upgrade Slots will be reset.
Once all the upgrades up to 135 RP are unlocked, use Research Builds to progress and get gems. Meanwhile, you will automatically gain the secret research trophies and upgrades.
At R23, research becomes available for the Neutral Factions. Unlocking them will require fewer steps as there is no new facility to unlock.
At R29, Prestige research becomes available. When unlocking Prestige research, do not buy the Vanilla unique building upgrade or the Prestige Trade Treaty will not show up. Only once unlocked can the Vanilla unique buildings can be bought before.