Can a Goblin use a Greataxe, or two Battleaxes? How much damage does he do with them?
I found a section (CRB p295) about the bulk and price differences of weapons of different sizes, but not about the damage differences.
Can a Goblin use a Greataxe, or two Battleaxes?
To expand on that, generally size of weapons and creatures has much less impact on the game than it did in PF1. To the extent that small and medium size has no impact on weapons - a small creature can wield a medium weapon, just as easily as a medium creature, and vice versa. In effect there is no such thing as a medium or small weapon, there are just normal sized weapons, which both medium and small creatures can use normally.
N.b. The Lost Omens Ancestry Guide, which comes out early next year will have a tiny playable race. But we will find out how that works when it is released.
How much damage does he do with them?
See List of weapons on Archives of Nethys, then assuming he hits, but doesn't critically hit;
With a Greataxe he would do 1d12+str slashing damage
With a Battleaxe he would do 1d8+str slashing damage
Outside of specific abilities, such as Double Slice, dual wielding only confers an advantage from providing the player choice. I.e. If you wield a Battleaxe and a club, you can choose which one to strike with. This is particularly useful for weapons with traits like agile, which reduce the multiple attack penalty, but are otherwise generally slightly weaker to compensate.
For example a rogue duel wielding a rapier and a short sword is a fairly common setup. The rapier is more powerful, but the short sword has agile.
CRB 295 from Archives of Nethys - Items and Sizes
Weapons sized for small and medium creatures are functionally the same size - a goblin can wield them with no penalty. A battleaxe sized for a large creature will be problematic for a small goblin to wield.
There is no longer any mechanical bonus just for using a weapon of a larger size in this edition of the game - not only is there no bonus, but you'll have a persistent clumsy 1 while you wield an oversized weapon.
In rare circumstances, some abilities will break this rule - like a giant instinct barbarian.