Even though the king's victory at Shrewsbury brings a distinctive sense of closure to the play, it's also fair to say the ending of Henry IV Part 1 has a major "to be continued" vibe. (In fact, the story does continue in the sequel, Henry IV Part 2, though it's not clear that was Shakespeare's original intention. Read "What's Up With The Title?" for more about this.)
Seems like we should have a little recap, no? The showdown with Hotspur's troops is over by the end of Henry IV Part 1, but Henry's got to wrangle up the stray rebels, who still pose a threat to the king. This is why Henry's final remarks in the play are orders for Prince Hal to head to Wales (to mop the floor with Glendower and Mortimer) and for Henry's other son, Prince John, to head up to York (to lay a fifteenth century-style smack-down on Northumberland, who phoned in "sick" instead of fighting at Shrewsbury).
In this respect, it's tempting to look at the ending and say, "This totally stinks. We want more closure! And more blood!" but there are some things we can take away from the play's conclusion.
We should point out that the play opens and closes with civil strife, which seems to gesture at the seemingly endless troubles of war and conflict in England. This point would not have been lost on Shakespeare's Elizabethan audience (it's not lost on us either). Two hundred years after the reign of Henry IV (when Shakespeare wrote the play), Queen Elizabeth was still dealing with the same kinds of issues (the Northern Rebellion, war with Spain, the Irish rebellion, and so on). Sorry. We didn't means to depress you. But, the thing is, war is depressing, you know?