He originally studied in Corfu under Charalambos Pachis (1872-1875) and completed his studies in Rome, Naples and Venice (1875-1877/78). Returning to Greece he settled in Corfu but also made many trips to Constantinople, Asia Minor, Egypt, Italy, France, Spain and Switzerland, painting landscapes. He began to exhibit in 1875, participating in the Olympia Exhibition, while in 1886 he presented his first solo show at the Athenian Club. There he met the British Ambassador Ford, who commissioned him to do seven albums with landscapes from Constantinople, Rhodes, Venice and Spain, organized his exhibitions in Athens and abroad and introduced him to European court circles. He continued to exhibit throughout his artistic career in both solo and group shows in Greece and abroad, among which were ones at the “Parnassos” and Zappeion Halls, the Artistic Exhibition of Athens in 1899, and the World Exhibition of Paris in 1900. In 1918 he had a large solo show at the GEO gallery, while in 1974 the National Gallery held a retrospective exhibition of his work. In 1902 he founded the Art School of Corfu, at which he also taught, and in 1907/8 he decorated the Vrailas house on Corfu, the country residence of Elizabeth the Empress of Austria, with wall paintings.
After experimenting with a wide range of expressive methods, even while still a student, Giallinas turned to water colour, which he cultivated almost exclusively and developed into one of the most outstanding Greek water colourists, creating a whole school. Painting mainly landscapes (seascapes, monuments of ancient Greece, Byzantium and the East, neighbourhoods and so on) but also a number of genre subjects, he created poetic images which reverberate with a classicistic-romantic point of view and are characterized by a balanced composition, showing special concern for the rendering of details and a sensitivity to chromatic nuances.