Active skills come from Uncut Skill Gems, which drop at area level. You 'engrave' an Uncut Skill Gem to pick or upgrade a specific skill.
Support gems modify a linked skill. As of 0.3.0 the one-copy-per-character restriction was removed, and supports now carry tiers (for example Brink I/II) that set their stat ranges. Higher support tiers are gated by your Uncut Support Gem level (1, 2 or 3).
There is no '5-tier' support system: the number 5 is the socket limit - the maximum number of support gems you can link to a single skill. You add support sockets with Jeweller's Orbs (Lesser -> 3, Greater -> 4, Perfect -> 5 sockets).
Leveling grants passive skill points and access to higher-level gems and gear.
Each base class unlocks ascendancy specialisations through the Trial of the Sekhemas and the Trial of Chaos.
Equipment provides defensive and offensive stats plus rune sockets; runes add fixed stats such as resistances or life.