The Ethiopian army was a mish-mash at the time.
First of all, you had the Kebur Zabagna, the Imperial Guard. This was a force trained by the Belgian military mission to about regular European infantry level. They used FN-Mausers, FN-BARs and Czechoslovak ZH-29 semi-automatic rifles. They also had heavy machine guns, a brigade of artillery with 40 French 75mm mle1897 field guns, at least (and probably more) 40 81mm mortars, a bunch of 20mm Oerlikon AA guns and 12 German 37mm Pak35 AT guns. The men were recruited among those loyal to the Emperor, but also men that had military experience. There were veterans from Italian Ascari units and British King's African Rifles among the Imperial Guard. The main problem for this about division-siezed force (after expansion at the outbreak of war) was that the Emperor relied on them to keep his internal enemies down, which made him reluctant to use them. They were not used in force until the last large battle of the war, at Mau Cei in March 1936.
Then you have the Mahel Safari, the traditional regular army of Ethiopia. These men carried pretty much what their fathers had carried at Adua 40 years earlier. Single shot and bolt action rifles were common, some in very high calibre and using black powder. They also had heavy machine guns, which they also used for AA purposes. They also had about 50 Russian 63,5mm mountain guns M1883, 56 ex-Italian 70mm M1885 mountain guns (captured at Adua 1896), 40 42mm Hotchkiss quick-firing guns and a smattering of Hotchkiss 37mm revolver cannons. There were also a smattering of old muzzle-loaded bronze artillery from the second half of the 19th century used.
Total number of artillery pieces in Ethiopia at the outbreak of war was reported as 234.
Thirdly, you have the Chitet, the feudal levies of the lords and governors of Ethiopia. They varied a LOT in quality and equipment. There were formations from the southwestern and southeastern part of the country where half the men carried spears and swords and shields exclusively. However, the formations in the northern and central part of the country often had exclusively rifles and some artillery and machine guns. The Swedish military mission in the north started the Oleta Cadet School and convinced the northern feudal levies to dye their traditional white garb tan for war and to use rudimentary infantry tactics (cross-fire, taking cover, flanking attacks, digging in on defence and so on). The Oleta Cadets marched out together with old men that arrived late in Spring 1936 and fought under Viking Tamm, a Swedish officer, in a battle before dispersing and joining the Abernyotch resistance.
The Ethiopians had 3 Ft 3000 tanks and captured about 18 L3/33 tankettes, at least one of which was used with the Army of Illub Abor in the Ogaden desert together with 6 MG-armed (and possibly armoured) trucks.
They also had a smattering of planes, mostly transports and trainers, but 6 Potez 25 recon/bombers were present. They were mostly used for recoinnasance and liason flights, as the supply of bombs was really low and their MGs had been removed earlier.
There was also a Turkish military mission training the mostly muslim feudal levies around the Harar region and a Swiss military mission trainding the Addis Ababa Gendarmerie.
The main problem for the Ethiopian forces were the lack of coordination and communication. Most orders went by horseback and the troops were slow to assemble and move. Supply was hard, especially as many weapons were outdated and food and other supplies moved mostly by mules. The Italians, enjoying total aereal superiority could attack the supply columns with impunity, often using mustard gas against them.
Many of the Ehtiopian officers and military governors were still fighting the old traditional Ethiopian way of charching right at the enemy and then retreating if he had not been broken after three days. The Ethiopian supply system was patchy at best, and the ability of the troops to stay in the field was limited. Several formations formed, fought, went home and formed again. The feudal levies of the sons of Ras Kassa reformed and attempted to recapture Addis Ababa in June and July 1936, arriving after the capitol had fallen to the Italians and the Emperor had fled.