The Polish Army’s original roots actually predate Polish independence by several years, as the Polish legion within the Austro-Hungarian army. Led by Jozeph Pilsudski, they participated in many skirmishes and battles during World War One. Many of these officers later formed the backbone of the Polish Army, but to assume that the Polish-Soviet war saw massive clashes of armed forces in early 1919 would be a misnomer, the Soviet Western Army wasn’t even formed until November 1918, and the Polish could only muster about 10,000 troops to occupy portions of the Ober Ost region (the undefined borderland between the Polish and Soviet Core areas). The war itself started on February 14th 1919, when Captain Mienicki captured the township of Bereza Kartuska with less than 60 troops, and captured 80 Soviet Prisoners. These extremely small troop conflicts formed the basis of the conflict, as neither Poland nor the USSR could actually muster troops.

On February 26th 1919 the Polish Sjem (Parliament) formally created the Polish army structure, mind you, this is weeks after the fighting and skirmishing had begun against the USSR. Up to this point, the remnants of the Polish Foreign Legion, whose allegiance to Jozeph Pilsudski was the only unifying factor, had essentially been defending the nation, and been skirmishing with the USSR. The total amount of forces at Poland’s disposal was roughly 110,000 troops on February 26th. Over roughly the next 6 months, other Polish contingents from World War 1 assembled and rapidly swelled the Polish numbers. The first Polish offensive had reached its zenith by April 21st when Pilsudski had occupied “Wilno” or Vilnius. There were many more skirmishes/offensives by the Polish in 1919, and essentially a rough border had been drawn (at least in practice) by September/October of 1919, and there was considerable talk of peace between the USSR and Poland, but that collapsed on November. This renewed conflict towards the end of 1919-1920 is really when the start of massed troops conflicts began in earnest.

In regards to the armament supplies questions, most of the war was fought on World War One overstock, and Poland did receive some aid from the Allied powers, but it was extremely limited. The UK turned a blind eye on Poland, and refused its requests on October 24th of 1919, but France did provide some material and financial aid, in the form of a 375 million frank loan, and a 50% discount on military purchases, and the United States provided a windfall of 56 million. However, in the grand scheme of Polish operations, these gestures, while beneficial, were mostly just that, gestures.

In conclusion, those grand armament and troop conflagrations didn’t occur until starting around March 1920, after more than a year of conflict.

There is a noticeable lack of English literature on the Polish Soviet War, the source material for my reply has been based off the historian Norman Davies “White Eagle, Red Star”.

/u/Tyler_E1864's answer above has already covered the general chronology, so I'd just like to supplement that with a bit more specific detail on where the troops were coming from.

While the core of the Polish army in 1920 was indeed the reorganised Polish Legions under Józef Piłsudski that had been part of the Austro-Hungarian and latterly the German armies, the Polish army as a whole was incredibly diverse in origin, which resulted in significant potential supply problems. The Polish army included elements that were formerly part of the Austro-Hungarian, German and Russian regular armies, and which therefore brought their own equipment; there was a 'Siberian Brigade' which had been formed around a group of Polish prisoners of war in Siberia, and which was partly armed with Japanese equipment; but one of the most substantial sub-sections of the Polish army was the so-called 'Blue Army' of Józef Haller, which was assembled in France (hence its field-blue uniforms) from a mixture of Polish POWs on the Western front and volunteers from the Polish diaspora, and included a small tank regiment – all of this was French equipment. There were plenty of volunteer contingents as well: for instance, much of the Siberian Brigade was made up of such volunteers, with the original core of 900 ex-POWs being elevated to officers and NCOs within the formation. Such volunteers also included an American fighter squadron, the 'Kosciusko Squadron', which contributed to the decisive aerial advantage held by the Poles in the conflict.

Essentially, the Poles did not raise an army from scratch. Rather, there was a process of agglomeration whereby various formations came together over the course of 1918-20 to form the army that fought against the Russian SFSR. Most of this army already had its own officers, training, equipment and organisation, and this did in the event produce some friction, not just logistically. In particular, some commanders, especially Haller, were more closely affiliated with the émigré nationalist movement of Roman Dmowski which became the National Democratic Party, rather than the more grassroots and, moreover, Commonwealth-restorationist movement headed by Piłsudski's Polish Socialist Party. Piłsudski's disastrous invasion of Ukraine in 1919 proved a severe hit to his political status, and at the height of the Soviet counter-offensive in July 1920, a State Defence Council which included several opposition party leaders was assembled in order to limit Piłsudski's autonomy. The next month, the civilian government began relocating to Poznán, which was one of the key areas of support for Dmowski and the National Democrats. These did not prove to be fatal to the Polish war effort in 1920, but the strain of factional infighting presaged the political violence that would ensue after Piłsudski's coup d'etat in 1926 and the formation of the Sanjaca dictatorship.