Uncle Edmure had sent Lord Vance and Lord Piper to hold the pass
At the beginning, Lannisters amassed a large host at Casterly Rock. Upon hearing about Lannister mobilization, Edmure Tully also called his banners and sent a small force Commanded by Lords Vance and Piper to guard the pass at Golden Tooth. This was necessary because Golden Tooth was last Lannister stronghold on River road which leads from Heart of Casterly Rock directly to Riverrun.
NOTE: To zoom the maps, you have to click on them.
On the map, you can see exact positions of all the hosts. At this time, Jaime is on his way to Casterly Rock after wounding Eddard Stark & Gregor Clegane is unleashed on Riverlands. The Mountain circumvents Lord Piper's Troops in Pass beneath golden tooth, crosses the red fork and massacres smallfolk in Sherrer and Mummer's ford. The Lannister main host however is still positioned at Casterly Rock.
the Kingslayer descended on them and put them to flight. Lord Vance
The Lannisters quitely followed the River Road North-Eastwards and descended on Piper's host below Golden Tooth. Jaime Lannister was commanding this host and he completely surprised his foe. The result was a crushing defeat for the Tullys.
Piper took the survivors and made to Riverrun to join Edmure's host but Jaime kept chasing him all the way to Riverrun.
All the time they were battling in the pass, Lord Tywin was bringing a
second Lannister army around from the south. It’s said to be even
larger than Jaime’s host.
Robb was situated in north at the time so Tywin's army was indeed south to him.
Tully's did not know that another Lannister Army commanded by Lord Tywin was already on its way to Riverlands via the Gold Road. Tywin struck Eastward on this road with the junction between Kingsroad and Gold road as his target.
As you can see in the image, the battle at the Golden Tooth and movement of Tywin's Army in the South, moving Eastwards.
Lord Derik had no sooner crossed the Red Fork than the Lannisters fell
upon him, the king’s banner be damned, and Gregor Clegane took them in
the rear as they tried to pull back cross the Mummer’s Ford.
By that time, Iron Throne had received the complaints of Riverlords about the raids conducted by the Mountain. Eddard Stark commanded Lord Beric Dondarrion to execute Ser Gregor Clegane in the name of King Robert I Baratheon. Eddard did not send Lord Beric to take on the entire military of Westerlands, that's Robb's wrong assumption.
Lord Beric rode from King's Landing and cut cross country into Riverlands in Western Direction. To enter the Westerlands (Domain of Lannisters), he needed to cross the Red Fork of trident river. He decided to make the crossing at Mummer's Ford. What awaited him there was an ambush. When he tried to fall back to Mummer's ford, Gregor Clegane blocked his retreat and attacked him from rear.
Afterwards, Jaime fell upon the Tully Troops assembled outside Riverrun. Another defeat for House Tully occurred. Jaime took Ser Edmure Tully as his Captive and laid siege to Riverrun.
All this While, Lord Tywin kept advancing towards Kingsroad in North-East.
You can see locations of battles of Riverrun and Mummer's Ford on the map. You can also see Tywin's Advance in Riverland-Crownlands axis from the South.
Lord Tywin has closed off the kingsroad, it’s said, and now he’s
marching north toward Harrenhal, burning as he goes.
As you can see in the Map, Tywin has now closed off Kingsroad for KL-bound traffic coming from North, Vale and Riverlands. His line of advance is showed by a red arrow. He was moving North towards Harrenhal. There he would move to capture the castle from Lady Shiela Whent. Then he will move further North to Lord Harroway's Town. There he will occupy the Ruby Ford and wait for the Starks & Arryns to come and try to relieve Tullys. Ruby Ford is the only crossing available if the Freys refuse anyone to use their bridge. Tywin had kinship with Walder Frey due to marriage between his sister and Frey's son. Also, Frey was a cautious man so Tywin assumed that Starks will try to force a crossing at Ruby's Ford. So as you can see, Tywin did not magically teleport to Ruby Ford. He had to March East and then North towards Harrenhal and then to Ruby Ford. The Cross-road inn was a strategic point which guarded the approach from Vale and North to Riverlands. It was to be expected that Arryns and Starks will come to aid of Hoster Tully as Lord Stark was Lord Tully's son in law and Lord Arryn was Lord Tully's grandson.
At the same time, Jaime completes siege of Riverrun from all three required directions by dividing his army into three camps.
If you wish to see graphical representation of events that followed these developments, you can see my previous answer here regarding Robb's advance, Battle at Green Fork, Battle of Whispering Woods & Battle of the Camps.