Sinatra didn't just "do sessions" with Tommy Dorsey from 1940-42: he was the band's regular vocalist, meaning he recorded with them, performed on radio with them, and toured with them. In 1942, he did his first "solo" recordings--four titles--while still with Dorsey. These recordings--including his first of "Night and Day"--are among the most beautiful he ever made.
Later that year, he left Dorsey to go solo, and signed with Columbia, where he recorded from 1943 until 1952. The best of these are from 1943-46. You could go with the four-disc set, which is an excellent selection up through '52, or go bonkers and buy the 12-disc box of all his Columbia recordings. Both have identical sound quality.
Of course as with most "complete" sets, you have to take the bad with the good. From, say, 1949-52, Sinatra was in generally poor voice. This was during the period where he lost his voice in a performance at the Copa and had to take several months off.
Although he later bounced back and made some very fine recordings for Capitol, to my ears his voice had lost that sweet, youthful quality for good. He was never quite the same again.
