I often get asked questions about Lucien's training system, so I thought I'd do a detailed breakdown of how it works here! Lucien is designed to feel like a "real person" and as such it's best for immersion not to get into the nitty-gritty of how he works, but for those who are super curious, here's a quick peek behind the curtain.
Phase One: Following Your Lead
Lucien is an observant sort of chap, and he's keen to learn as much as he can, whenever he gets the chance. From the moment you first agree to take him with you, he'll begin studying your techniques and improving his own skills based on yours.
What this means is that every 3 days Lucien travels with you, he'll gain one skill point in each of your skills that's a higher level than his. For Health, Stamina and Magicka, this is done in increments of 5. For example:
If you have 230 Magicka and a One-Handed skill of 45, and Lucien has 100 Magicka and a One-Handed skill of 10, then after 3 days, he'll increase to 105 Magicka and a One-Handed skill of 11. It'll continue like this every 3 days until he catches up with you. Make sense?
Phase Two: Learning On His Own
Lucien doesn't wholly depend on you - he also learns by doing! Every 3 days, after the phase one skill increase is applied, Lucien gets 3 more skill points, which are randomly distributed among skills relevant to the combat style you have chosen for him. Health, Stamina and Magicka are also in the mix, once again in increments of 5.
So, if you've told Lucien to fight as a mage, he might get a point in Destruction, a point in Restoration, and 5 points in Magicka. Or he might get 3 points in Alteration, or 15 points in Magicka. What he'll never get, though, are points in Stamina, Health, One-Handed, Archery, etc. etc. unless you choose a different combat style for him. If you tell him to fight however he likes with no particular combat style, these points could be in any follower-useable skill.
The idea with this is that you can have a mage Lucien even if your character is a broadsword-wielding barbarian. He won't progress as fast as he would if you yourself had some skill in magic, because then you'd get the Phase One bonuses as well as the Phase 2, but he'll still gradually progress.
Phase Three: Life Lessons
Once he has travelled with you for a little while, and Lucien has become a bit braver and/or started to trust you, he'll ask you to train him. As I mentioned above, my recommendation is to let this happen organically rather than deliberately forcing it to trigger, but the best way to earn his trust is to ask him about himself and do quests with him that he recognises.
If you agree to train him, you'll unlock a new dialogue option: "I'd like to start your training." Now, once every 24 in-game hours, you can give Lucien a lesson in any skill you have that's 5 levels or more higher than his equivalent skill level. The screen will fade to black, you'll hear a snippet of him trying to learn, an in-game hour will pass, and when the screen fades back in, he'll have gained permanent 5-point boost in that skill.
So if he has an Archery skill of 30 and yours is 48, you'll get a dialogue option to train him in that skill, which will increase him to 35. On the other hand, if your Archery skill is only 34, no option will appear to train him in Archery.
You can also choose to teach him a new spell instead of a particular skill boost. This option is available if you have one of the relevant spell tomes in your inventory. You don't have to know it yourself. Lucien can learn any of the spells from this list, so have a gander at that to see you can teach him.
Hope that all makes sense! Drop me a comment if anything's unclear, and look after your Luciens! :)