That's because Robert's allies were up north and he did not want to get surrounded by Armies of Crownlands, Highgarden and Dorne.
Geography of Westeros
In Westeros there are nine administrative units which can be seen here:
As you can see, Stormlands are surrounded:
- In North, there are Royal fiefs of Crownlands.
- In East there is the Narrow Sea.
- In West, There is the Reach.
- In South, There is Dorne.
When the War broke out, Following was the Breakdown of the forces:
- the North
- the Vale
- the Riverlands
- the Stormlands
As you can see, All of Robert's allies were up North. North, Vale and Riverlands shared borders with each other with only Riverlands sharing borders with Loyalist territories.
Stormlands alone were however surrounded by the Loyalists on every compass point. So Robert wanted to advance his forces north and link up with his allies before he could be trapped between the loyalist forces of Dorne, Reach and Crownlands. But he had to first deal with Storm Lords who were trying to rise up for the King.
Robert had won victories at Summerhall against those Lords and now had the complete support of Stormlands behind him. With home front secured, Robert advanced North-Westwards to link up with his allies; The Houses Tully, Arryn and Stark.
Naturally, to reach Riverlands, Robert had to either cross the Crownlands or the Reach. Robert chose to cross through the Reach and came upon the border town of Ashford. It is unclear if Robert tried to take Ashford castle before advancing to Riverlands but we know he was soon faced by Tyrells. Lord Mace had bestirred himself and Lord Randyll Tarly, known as finest soldier in the realm, had reached Ashford with his vanguard.
Tarly did not wait for main Tyrell Army to arrive and charged at the Stormlanders. Stormlanders suffered losses and on arrival of main Tyrell Army, Robert was forced to withdraw to Riverlands. This is the only defeat Robert ever suffered.
That's how Robert reached Riverlands and the reason why further confrontations happened in that region as:
- Riverlands were the only point of entry into Rebel territory
- Rebel leaders, the Lords Arryn, Stark, Tully and Baratheon were present in Riverlands.
How did Rhaegar reach Riverlands from Dorne?
Rhaegar had gone to Dorne, that's correct.
However, the Loyalists suffered a very heavy defeat in Battle of the Bells. That's when Rhaegar returned from Dorne and took command of the remnants of the Royal Armies, regrouped and reinforced.
From ASOS: Jaime V:
He floated in heat, in memory. "After dancing griffins lost the
Battle of the Bells, Aerys exiled him." Why am I telling this absurd
ugly child? "He had finally realized that Robert was no mere outlaw
lord to be crushed at whim, but the greatest threat House Targaryen
had faced since Daemon Blackfyre. The king reminded Lewyn Martell
gracelessly that he held Elia and sent him to take command of the
ten thousand Dornishmen coming up the kingsroad. Jon Darry and
Barristan Selmy rode to Stoney Sept to rally what they could of
griffins' men, and Prince Rhaegar returned from the south and
persuaded his father to swallow his pride and summon my father.
He then marched towards North and tried to cross the River trident.
Rebels decided to stop the advance of the Prince at River Trident because given the terrain of Riverlands (All plains and rivers) that was the nearest defensible position against the Advancing Royal Armies. A battle ensued which decided the fate of Targaryen monarchy and the Crown Prince was killed.