It doesn't seem like there'd be a job interview process, so how did the condottieri recruit? Did people supply themselves with weapons, horses, food, fodder, etc.? Were there certain types of soldiers that were in-demand?
A lot of Condottieri we're generally men who had previously fought as a soldier before becoming a mercenary.
One of the most famous mercenaries in Italian history was actually an English man, John Hawkwood, who fought initially in the hundred years war. There is some debate of his background, but he was probably the son of a tanner who was recruited or joined the army, after a short apprenticeship as a tailor. After the treaty of Calais, which ended the initial phase of the hundred years war, Hawkwood ended up as a mercenary, eventually gaining command of the White Company, a relatively famous mixed nationality mercenary force. Hawkwood fought initially for the Pope, but changed his allegiance quite regularly, until later in the 1390s, when Florence offered Hawkwood command of the Florentine army against Milan, he successfully defended the city from the Milanese, and was granted citizenship, and a pension. He rarely fought again, his contract with the city meant he could not fight for anyone else.
After his death, the city of Florence commissioned a famous painting of Hawkwood as if part of an equestrian monument, within the Cathedral. A pretty unheard of honour for a mercenary.
The source I found most helpful is a recent publication John Hawkwood: An English Mercenary in Fourteenth-century Italy, written by William Caferro
[There's also this english translation of a much older narrative of Hawkwood's life, written by Giuseppe Marcotti] {https://archive.org/stream/sirjohnhawkwoodl00lead#page/n9/mode/2up}