During this early, formative time frame - and following the defeat of the Spanish - centralist 'Unitarios' waged a lengthy conflict against Federalists to determine the future of the nation.
The dominant figure of this period was the Federalist Juan Manuel de Rosas, generally held to be a tyrant, who ruled the Buenos Aires province from 1829 to 1852.
Rosas developed a paramilitary force of its own, La Mazorca ("the Corncob"), which earned the federalists the derogatory nickname of mazorqueros, even if they preferred to be known as The Holy Federation...
After a revolution under General Justo Jose Urquiza, a defecting Federalist supported by Uruguay and Brazil, Argentine national unity was at least nominally established, and a constitution promulgated in 1853.
During the time frame from 1865–1870 - the bloody 'War of the Triple Alliance' was fought by Argentina, Brazil, and Uruguay against Paraguay.
Following that - Argentina's military would deal with many and various political intrigues and fight the small groups of 'insurrectos' while maintaining close ties to the foreign money that controlled the nation state through heavy investment.
The years from 1880 to 1900 and beyond brought Argentina intensified economic prosperity, mainly by way of trade with Europe, and the military heirarchy was closely aligned with the Argentine upper class - especially the great land owners - who expected the maintenance of their status quo.
Hasn't changed all that much...
You can draw some more from the 'Area Handbook - Argentina' - and no doubt other sources - but beyond a thumbnail sketch , little will be available on Argentina's early military involvements - and since she was a neutral nation state - there weren't any troops committed to anything outside the contiguous borders.
Apparently, squabbling with her neighbors and keeping the rural folk in line by whatever means necessary kept them busy enough.
Like many small countries - Argentina was interested in more efficient ways to keep her thumb down on the populace where needed and to deal with her enemies when pointed out - so she looked to both Europe and America for ideas.
Arms merchants everywhere found Latin America a fertile ground for sales and testing - as well as a dumping ground for obsolete weaponry no longer issued.
Good Luck.
Vaya,
Scouts Out!