This is information which is less than clear from reading the Throne of Bhaal manual.
A character gains his/her first high-level ability as soon as the character levels up at or above 3,000,000 experience points.
For a dualclass character (a character who began with one class and then switched to another one), the first high-level ability (which must come from the second class's pool) is gained when the second class levels up with at least 3,000,000 experience points in that class specifically. Experience points in the first class do not matter.
For a multiclass character (a character who simultaneously advances in two or three classes), the first high-level ability is gained as soon as either class levels up when the /character/ has a combined total of
3,000,000 xp. High-level abilities for such a character can be selected from a combined pool of the high-level abilities for two classes (whether the character actually has two classes or three). A character with three classes will not have access to the mage pool. Thus, a cleric/ranger can choose his/her first high-level ability upon reaching level 13 as a ranger, will gain another high-level ability at every level thereafter, and is free to choose from the cleric pool or the ranger pool regardless of which class has actually leveled up.
High-level ability pools: Paladins, regardless of kit, have access to the Summon Deva ability. Each time they select that ability, they gain the innate ability to summon one deva, as per the spell, once per day.
Thieves with the Swashbuckler kit do not gain access to the Assassination ability. Instead they gain access to the Whirlwind Attack ability, though none of the other warrior abilities.
Rangers gain access to the Tracking ability. This is stated at one part of the manual but contradicted in another.
The Critical Strike ability can be taken any number of times, once the character has the Power Attack prerequisite once.
The Use Any Item ability overrides alignment, class, and name restrictions, but not stat restrictions (a Chaotic Good thief with Constitution 17 and Use Any Item can use the Holy Avenger sword which is restricted to paladins, or the Soul Reaver sword which cannot be used by any Good character, or the Human Skin armor which can only be used by Evil characters, but will not be able to use the Sword of Chaos +4, which requires its wielder to have Constitution 18). Bear in mind that most items designed for a specific NPC are only usable by a character who equals or exceeds several of that NPC's stats. However, if using an item imposes a restriction on a certain class, that restriction will remain in place even with Use Any Item. (A thief can use weapons normally not available to thieves, but cannot backstab with them, and a thief/wizard can wear any armor, but cannot sneak wearing armor heavier than leather or cast spells wearing any armor.)
The Elemental Transformation abilities from the druid pool are innate abilities, not spells. The druid can change into an elemental once per day for each time the ability is selected.
Fighter and thief high-level abilities are considered innate spells, and thus any number of them can be used with Improved Alacrity.