How was (non-instantaneous) electric current first discovered and what were some of the main first thoughts on it?
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I will assume "non-instantaneous" means something other than electric discharge in the atmosphere, from animals like eels and torpedo fishes, or electrostatic generators like the Leyden jar or the van der Graaf generator. Then the answer is the voltaic pile invented by Volta in 1799, "the first electrical battery that could continuously provide an electric current to a circuit". The invention was inspired by Galvani's experiments with animal electricity, but Galvani himself used electric sparks. Since prior to Volta only static electricity was dealt with there was a controversy at to explaining phenomena with the newly discovered steady current. Despite Volta's own intuitions about the electric fluid propelled into "perpetual impulsion" by an "electromotive contact force", the original theory of the pile, adopted by an illustrious committee of Laplace, Coulomb, Fourcroy, Guyton, and Biot, was quasi-static, the current was interpreted as a series of electrostatic discharges. Science getting decided by committee is rarely a good sign.
The means to measure current did not appear until Oersted's observation of its effect on magnetic needles. Ampere, an outsider to the French scientific establishment, dismissed Biot's quai-static theory in 1820-s, and reintroduced Volta's original intuitions about perpetual circular movement of the two electric fluids in the wire propelled by the pile, "reunions in the wire" as he called them. But even Ampere's theory was intermediary, it sharply separated electrostatic effects in the pile, and electrodynamic ones in the wire, tension electricity and current electricity, as they were called. Only in 1827, when Ohm formulated his law, the modern definition and theory of the electric current emerged. Ohm's law required a reconceptualization of electricity that was seen as radical at the time, and met much resistance. To his contemporaries, his now familiar law looked like a formula linking the intensity of light to its frequency, a nonsensical connection. For more details see What is the history of electric current and resistance?