Decide if you wish to collect all of Sinatra's recordings; assemble a conoisseur's collection; or collect only his best music. This will determine if you buy, for example, "Best Of" and boxed sets, versus assembling the albums one by one.
Practically all of the albums are available on CD format, as are complete collections from significant periods of his career (covering the Columbia Records years, Capitol Records years and Reprise years).
Purchase the Capitol Records albums, generally considered his best work, recorded between 1953 and 1960. During this time, he sang such enduring hits as "Witchcraft," "All the Way" and "Learnin' the Blues."
This period was the heyday of Sinatra's career. During this time, he pioneered the "concept album," an album of songs with a single theme; these included "No One Cares," "Sinatra Sings for Only the Lonely," and so on. These were the years when he entered motion pictures (winning an Academy Award for "From Here to Eternity"). These were also the years during which Sinatra married Ava Gardener and established the "Rat Pack" (including Sinatra, Dean Martin, Sammy Davis Jr., Joey Bishop and Peter Lawford). Finally, this was the time when Sinatra became involved in the election of John F. Kennedy.
A good single source from this period is "The Best of the Capitol Years," released in 1960.
A more comprehensive source is the four-CD box set, "The Complete Capitols Singles Collection."
Finally, if you wish to collect all Sinatra, work backwards from 1960, running through his discography (See Reference 2).
Purchase the Reprise records. Sinatra established this label in 1960, and sold it to Warner Brothers in 1963, but maintained artistic control. These recordings are not considered quite as good as the Capitol recordings, but were entirely Sinatra's choice, and his favorites. He described himself as a "New, happier, emancipated Sinatra, untrammeled, unfettered, unconfined" (See Reference 2).
Historically, this period was coincident with the height of the Rat Pack popularity, his estrangement from Kennedy, and his "Ocean's 13" film. During this period, he licensed and re-recorded such earlier favorites as "I've Got You Under My Skin," "Nancy," and so on.
There is no single, definitive compilation album from the Reprise years; some of the most significant albums from this period include "My Way," "September of my Years," and "Sinatra-Basie." "Ring-a-Ding-Ding" was the first, and a Sinatra favorite; "Ring-a-Ding-Ding" would become his catchphrase during the Rat Pack years.
Finally, there is an excellent four-CD set, "Frank Sinatra, the Reprise Collection," which includes every significant recording from this period.
Purchase his earlier work, including his stints with Tommy Dorsey (beginning in 1940), and his Columbia years (beginning the late 1942). This work included "I'll Never Smile Again," "All the Way" and such standards.
Columbia released four decent compilations of Sinatra's music: "The Best Of The Columbia Years (1943-1952)," re-released as a four-CD set; "The Columbia Years 1943-1952 The Complete Recordings"; "Portrait Of Sinatra: Columbia Classics"; and "The Best Of The Columbia Years...".
The most significant single albums from this period, if you choose to go that route, include "Swing and Dance with Frank Sinatra" and "The Voice of Frank Sinatra."
Fill in the blanks from the rest of his career. Such later works as "Old Blue Eyes is Back" (1973), "Duets" (1995) and the four-CD set "Sinatra At the Movies" and so on, are at times wonderful curiosities; oftentimes painful. But they fill in a complete collection.