Stats Areas

Aerial Magic
To Advance: Fly (Sky only). Data by Murtagh:
46 flights for Capable
55 flights for Effective

Alchemy
To Advance: Alchemize. Desert alchemy is universally available, although only Metal and Earth can utilize it. Metal souls have access to more knowledges. The only way for Leaf, Water and Sky to reach Alchemy '1' is to use the knowledges given by the spell "Alchemy." The most time-efficient way of gaining Alchemy skills is by using Adamantine Alchemy. This, however, is not available except to Metal souls and is somewhat costly in terms of items.
40 'Alchemies' are needed to attain Alchemy 1.
42 Desert Alchemies are needed to attain Alchemy 1.
108 Desert Alchemies are needed to go from Alchemy 0 to Alchemy 2.

Angelic Magic
To advance: Summon Souls, Ascend, Escape, etc. Very SE costly.
Angelic is the only area in which the gain of multiple stats through one action is recorded. Pakaran reports gaining both Angelic 2 and Angelic 3 from a single Ascend.

Astral Magic
To Advance: Sleep, Drift, Wake (Sky Only)
When going into Astral, be sure to queue a Wake!

Baking
To advance: Bake

Conjuration
To Advance: Conjure
Advancing Conjuration level is not directly related to the amount of SE spent.

Construction
To Advance: Build, Dig, Raze and Destroy

Death
To Advance: Die. Earth Soul with TD will be able to gain Death Stats
more easily because Earth soul can die without being dead or losing stats. Since Dying costs xp, sometimes to die is to regress in Death stats (lower or losing the area.)

Diplomacy
To Advance: Convert hamlets.

^^Elemental Magic^
To Advance: Practice Elementally specific magic. The most common way to gain xp is to raise/lower land. Charge Earth gives a boost to Elemental Magic level.

Gardening
To Advance:

Gleaning
To Advance: Glean. Current data shows (Thanks to blobglob, evyndd,
aeneas, and others):
Start: 2h 11m
After 40 gleans: 2h 8m*, Capable
After 80-82 gleans: 2h 4m
[After 95-99 gleans: Effective]
After 132-138 gleans: 2h
As of 171 gleans: still 2h, Effective
[Edit: After 173 gleans: Accomplished]

Hypnology
To Advance: Utilize Hypnology skills.

Illudomacy
To Advance:

Illusion
To Advance: Cast illusion related spells: Inficio Mundus, Temporvoltus,
and Devestus

Journey
To Advance: Walk a lot. The more people and the more weight gives
more stats.

Knowledge
To Advance: Teach, Learn, and Invent. Successful invents boost one's
knowledge level. According to Rabek's data: You lose a bit of your knowledge stat, as well, whenever you forget. You also gain knowledge even on failed invents. The amount that forgetting drains your knowledge is (as far as I know) equal to about one failed invent.

Leaf Magic
To Advance: Enchant or sleep forests. Leaf specific.

Limbology
To Advance: Forget knowledges or use limbology.

Manufacturing
To Advance: manufacturing alchemical instruments or other alchemical gadgets

Napping
To Advance: Wait.

Nourishing
To Advance: Casting Nourish

Sorcery
To Advance: Practice Sorcery such as Force Bolting (someone else or self)

Spatial Magic
To Advance: Charge Space, Trigger, Gather. Successful Gather seems
to give the most boost. A sky soul can gain a boost in XP by Aligning
Weir Gates.

Telepathy
To Advance: Utilize Telepathic powers. Creating and charging forums
seems to give the most boost. Probe and Scry give almost equal XP, according to JWRTolkien's data, but Scan gives more. The difference is between 1 minute/1 SE for Scry and 52 minutes/no SE for Probe. Excessive signalling causes telepathy exp loss.

War
To Advance: Engage in warfare. Killing large hamlets boosts War XP.

Weather
To Advance: Calling rain and summoning ocean current (summoning ocean current is only for water).