Where does Tyrion start?

Tyrion begins the series as the family embarrassment — the youngest Lannister, a dwarf, the one Tywin barely tolerates and Cersei openly despises. He drinks, visits brothels, reads obsessively. He's treated as a permanent liability. His intelligence is acknowledged in the abstract; he's not trusted with real responsibility.

How does he first prove his value?

At the Battle of the Blackwater. Acting as Hand of the King while Joffrey hides, Tyrion organizes the city's defense using wildfire, chain tactics, and morale management under fire. He leads soldiers personally. He nearly dies when Joffrey's men attempt to kill him during the chaos. He survives; Joffrey gets the credit. This pattern — Tyrion solving the problem, someone else claiming the victory — defines his arc.

What does his imprisonment and trial reveal?

That the Lannister system was never going to give him justice. He's accused of poisoning Joffrey (he didn't). The trial is theater. His witness list is a character assassination. He finally cracks when Shae testifies against him — and delivers his trial speech, formally renouncing the game he's been trying to win. He demands trial by combat.

What happens after his escape?

He kills Tywin, escapes to Essos, and eventually finds Daenerys. Serving as her Hand represents his highest position and his greatest failure — he consistently underestimates Cersei, mismanages Dany's campaigns, and ultimately can't prevent the destruction of King's Landing.

How does his arc end?

He survives. He talks the lords of Westeros into selecting Bran as king, makes himself Hand again. It's ambiguous — a victory with an asterisk. He ends where he started: the most clever person in the room, still not quite in control.