The lengthy title for the 1598 printing was "The History of Henrie the Fourth, With the Battell at Shrewsburie, between the King and Lord Henry Percy, surnamed Henrie Hotspur of the North, with the humorous conceits of Sir John Falstaffe".
Surprisingly, Hal, Prince of Wales, (later Henry V) was not even mentioned in this verbose title although many would consider him to be the central character. This play is clearly the dramatization of a struggle for a kingdom, but it is equally the story of Hal's wild and reckless youthful adventures with Falstaff and other disreputable companions.
Shakespeare did not write his plays about English kings in chronological order, but these plays do have a historical unity. It is helpful (but not essential) to read the tetralogy Richard II, Henry IV Part 1 and 2, and Henry V in chronological order. Whatever route you take, I highly recommend buying a companion copy of Peter Saccio's "Shakespeare's English Kings", an engaging look at how Shakespeare revised history to achieve dramatic effect.
A wide selection of Henry IV editions are available, including older editions in used bookstores. I am familiar with a few and have personal favorites:
The New Folger Library Shakespeare is my first choice among the inexpensive editions of Henry IV. "New" replaces the prior version in use for 35 years. It uses "facing page" format with scene summaries, explanations for rare and archaic words and expressions, and Elizabethan drawings located on the left page; the Henry IV text is on the right. I particularly liked the section on "Reading Shakespeare's Language in Henry IV" and Alexander Legget's literary analysis (save this until you have read the play). The fascinating article "Historical Background: Sir John Falstaff and Sir John Oldcastle" adds a religious dimension to the play that I had not previously noted.
The Bedford Shakespeare Series provides an excellent study text (edited by Barbara Hodgdon) titled "The First Part of King Henry the Fourth". It is a little more expensive, is about 400 pages, and provides a broad range of source and context documentation. It would be excellent for an upper level course in Shakespeare. The context documentation is fascinating and informative; it ranges from the Holinshed Chronicles to Elizabethan writing on Civic Order to detailed cultural studies of London's diverse populous. Other chapters address the OldCastle controversy and the "Education of a Prince".
I also like the Norton Critical edition (edited by James Sanderson), "Henry the Fourth, Part 1", particularly for its extensive collection of literary criticism. The essays are divided into two parts: 1) the theme, characters, structure, and style of the play and 2) a wide variety of interpretation directed toward that roguish character, Sir John Falstaff.