Story points should be rewarded for essentially any effort the players makes to drive the story forward. In many other traditional tabletop roleplaying games, the analogue to this progress system is typically rewarded for slaying creatures or completing a "campaign". You'll note that the Bestiary does not include suggested story points for each entry - this isn't to say that story points shouldn't be rewarded for slaying a great beast that threatens the town, but rather that the amount rewarded is intended to be flexible.
Story points are intended to be rewarded for:
All of the above actions (and more at the narrator's discretion) should reward SP (to be handed out when the encounter ends). The duration of any given encounter is collectively up to the players and the narrator, however the loose expectation is that any given play session will result in an average of 3 encounters. The narrator should determine the pace at which they wish players to improve - if a narrator wishes players to rapidly gain new abilities, they could consider handing out up to 10SP per encounter; the intended design was roughly 5SP per encounter (or, a total of 15SP per play session). In such a way, most players would be able to make one solid improvement on their character, on average, per session.
The slow and steady trickle of constant improvements is directly contrary to the "level up" system more commonly used, where enormous improvements in character quality come at once, but only after a prolonged period of time.
Ultimately, how much SP any encounter rewards is up to the narrator, and they will effectively set the pace of character improvement in this way. The game is designed for players to feel like their characters are constantly improving one step at a time; the narrator should be sure to read the players reactions - this is a game after all, and everyone should be having fun; if more rewards are needed, then perhaps that should be considered.
Loot comes in two flavors - props and specific rewards the narrator has baked into their story (the key to the lost keep, a promised bounty for an apprehended criminal, etc.) and "Loot" (randomly found items that players have the opportunity to roll for).
The Loot tables are intended to be used for two purposes; first, each Bestiary entry has a suggested set of Loot tables to roll on. These are only suggestions, and are not meant to be a hard and fast rule that players can use to force rewards from the narrator. The suggestions shown are thematic in nature - in this world, a certain beast would be expected to horde or have possession of certain items, and the Loot tables reflect that.
The second usage of Loot tables is as the narrator finds the need in the course of the story. Perhaps the players break into a bank vault in the middle of the night - the King's Ransom loot table provides the narrator a ready made and reasonable range of crowns that might be found in such a place. By using the loot table, the narrator again puts the action in the hands of the players (see PCs vs NPCs), which is an overarching theme of gameplay in the Twilight Kingdoms.
Loot that is considered a prop should be handed out as the story dictates. Loot that is a random reward should be handed out when it "feels good or right" to do so. Defeating a powerful foe feels like a good time to hand out loot, hence the suggested loot tables in the bestiary. Finding a secret stash or hideout also seems like an appropriate place to allow for the Loot action.
How much loot to allow is a matter of preference for gameplay pacing. Ingredients can be used to augment and empower players, so simply dumping tons of faecorns on them may not be a good plan. Be aware of any tales that players have taken that require certain loot to function (maybe faecorns should show up more often than normal if one of the players is very excited about their Alchemist tale; it is never a good idea to let a player select one of their three tales as a crafting tale, and then starve them of their needed ingredients.
Be aware that the Treasure table essentially can allow for anything in the game to show up. If the narrator wants to limit powerful things such as Tarr'ko and Devices early on (likely a good idea), then they should not take the easy Treasure table roll approach, and select narrower tables for players to roll.
Also be aware of how much luck any given player has, and if they have modifiers for rolling. If a player has chosen Sciura as their species, or has taken the Scoundrel tale, they have invested in being able to roll for loot, so always using prop rewards will be a disappointment to them.
Keep in mind that both Tarr'ko and Devices are intended to be end-game character improvements. These items should be very hard to find, and certainly shouldn't be available early on (unless you wish to start a game with powerful characters). It is intended that once a character has exhausted their affordable SP upgrades, they will enter into a phase of character improvement where a single SP upgrade comes slower, but is augmented at the same time by the occasional improvement in high-end item rewards (such as Tarr'ko and Spiretech).
Finding these items should feel unique and special, and should make a character excited to try out a new playstyle; keeping these as sparing rewards will allow them to continue feeling that way throughout an extended campaign.