After Tyrion descends from Vale and returns to the Riverlands with Bronn and the hill tribes, how does Tyrion know where to meet with the Lannister host and Tywin?
I'm not entirely sure if Tyrion set out specifically to find his father's host, but he did want to leave the Vale and head elsewhere. To do that his travels would ultimately take him to one familiar place, The Crossroads Inn. This inn lies in the crossroads between the Kingsroad (crossing the Seven Kingdom north to south), the River Road (east to west) and the High Road (the only major road connecting the Vale with the rest of the Seven Kingdoms). It was at this inn that he was first captured, and going to anywhere in the Seven Kingdoms coming from the Vale necessitates a stop there.
When he does get there, it just so happens that Tywin Lannister had made it his campaign headquarters, probably to punish the owners in their role in Tyrion's capture. So whether Tyrion was specifically looking for his father there or not might not be entirely clear, what is clear is that since Tywin was camped at the Crossroads Inn then Tyrion was bound to meet him.
When Tyrion got captured, Tywin marched against the Eyrie. Tywin's army was quite large, and the only way to move it was through the Kingsroad.
So Tyrion just set out south to go back to King's Landing and he fell upon his father's army.
I don't remember reading/seeing anything else. It was just a coincidence.
First of all you need to know about the road he took. When Tyrion left the Vale, he took the High Road which leads from Eyrie to Gates of the Moon. From there, it goes to Inn at Cross Roads which is situated at, well Cross roads where the High Road, River Road and King's Road meet.
So inevitably, Tyrion eventually reached the Inn at the Cross Roads. Luckily for him, his father was already there.
Why was Tywin Lannister at Cross Roads?
At that time, Eddard Stark was captive of Lannisters at King's Landing. Simultaneously, Jaime was advancing in Riverlands and had besieged Riverrun. Lord Tywin suspected that Vale, North and Riverlands will respond to arrest of Eddard Stark because:
- Eddard Stark's wife Catelyn Tully and Regent of Vale Lysa Tully were sisters.
- Lord Paramount of Riverlands Hoster Tully was father of both Catelyn and Lysa Tully.
- North was sworn to serve the Starks of Winterfell.
Striking first, Tywin Lannister sent Jaime to force Riverrun into submission before they could join their forces with Starks and Arryns. Then to protect Jaime's besieging force, He crossed the Green fork river and made camp at Inn at the Cross Roads, where King's Road from the North, High Road from the Vale and River road from the south meet.
He strategically situated himself there because any relief force from North or Vale would have to pass through there to enter Riverlands. Moreover he could stop any force which could threaten King's Landing from there. To see detailed commentary on movements of Jaime and Tywin in this regard, see my previous answer here.
Did Tyrion know about his father's movements?
No he did not. Tyrion had no idea that he might find his father on the road and neither that the war had started.
When Tyrion was on the High Road, he told Bronn that they were riding for the River Green fork. Then the sellsword asked him that what would he do when he reaches the river. From AGOT, Tyrion VI:
“If we do reach the river, what will you do then?” the sellsword asked as he cut.
“Oh, a whore and a featherbed and a flagon of wine, for a start.” Tyrion held out his trencher, and Bronn filled it with meat. “And then to Casterly Rock or King’s Landing, I think. I have some questions that want answering, concerning a certain dagger.”
As you can see, Tyrion had no idea that his father would be encamped at the river. He was just thinking about going to Casterly Rock or King's Landing because:
- From Cross Roads, he could take the River road Westwards, ride till Golden Hills, cross the Pass of Golden Tooth, follow the road along the sea and reach Casterly Rock.
- Or he could just take the King's Road to the South and reach King's Landing.
He did not know that Riverrun was under siege or that his father was encamped at the cross road. Then Tyrion had that whole episode with Clansmen and used them to scout the way ahead for him. Those scouts told him that a Lannister Army was encamped on the river. Even then he wasn't sure whose army might that be.
From AGOT Tyrion VII:
“By their fires I call them twenty thousand strong,” she said. “Their banners are red, with a golden lion.”
“Your father?” Bronn asked.
“Or my brother Jaime,” Tyrion said. “We shall know soon enough.”
Then it was Ser Flement Brax who informed Tyrion that it was indeed his Father's army in the same chapter:
Ser Flement Brax lifted his visor. “Tyrion,” he said in astonishment. “My lord, we all feared you dead, or...” He looked at the clansmen uncertainly. “These... companions of yours...”
“Bosom friends and loyal retainers,” Tyrion said. “Where will I find my lord father?”
“He has taken the inn at the crossroads for his quarters.”
TL;DR, Tyrion did not know where to find his father. He just followed the regular road, unbeknownst to him that while he was a captive, Robert Baratheon had died and War of the Five Kings had begun. Even until reaching the army itself, Tyrion wasn't sure that his father would be there. However, he was sure that there was a Lannister Army located at the cross roads thanks to scouting of Chella.