Allow me to use fashion as an analogy. Some trumpet players view their horn like a favorite coat—they wear the same one for everything, regardless of the event. Others change their horns around like celebrities change outfits, sometimes using several different ones throughout the day. Okay, I never claimed to be good at analogies.
The point is, you may or may not eventually see a need to own multiple trumpets. There are two versions of this need.
Multiple Standard Trumpets
Many players have various trumpets they use for different styles of music or for different specific needs based on the music they are performing. The untrained ear may not hear a difference, but the player may feel a whole world of difference! A wider bore or heavier metal horn may offer a trumpet player a darker, warmer sound to utilize for a ballad, whereas the opposite horn may offer some much needed brightness and volume. Within a normal set of music, these two separate needs would not be uncommon, and there are players that find switching horns during a single performance is almost as useful as switching mouthpieces. However, most multiple-horn-owning players simply have different horns they use for particular gigs—one horn for the classical wedding performance, one horn for the Salsa gig. It takes a long time, even years, and a lot of trying to find the right horns with the appropriate variations that will fit your specific needs.
If you are playing classical music, a piccolo trumpet may be a must-have for you. It is definitely its own beast and will require extra practice effort from you, but it is a highly important sound within the classical brass lexicon (and, occasionally even in pop music, most notably The Beatles "Penny Lane"). Usually, the musical parts the piccolo trumpet player gets to play can be pretty fun, albeit challenging. I have even seen some lead players use it to "cheat" a little on nailing some high notes.
C Trumpets (and Other Keys)
Again, these are horns you may only find yourself needing if you are playing classical or orchestral music. They do not often find their way into most commercial music styles, like jazz and pop. If you are looking at a career in the classical field, you probably need to start looking at various models of these C trumpets—you certainly will want to have them handy when working opportunities arise.
The cornet is, sadly, nearly forgotten in modern music. While the early masters made their marks on the cornet (King Oliver, Louis Armstrong, Buddy Bolden, etc.) and paved the way for every trumpet player to follow, a scant few players even own the a cornet these days. You'll most often find cornets played by musicians performing regularly in Dixieland bands or the occasional traditional marching band. Even in these situations, players often just use their regular trumpets, since the horns are so close in sound and almost no once can tell the difference.
But there is a difference. Every variation of a horn requires some level of adjustment in the player's approach, and to simply play a cornet like a trumpet robs the horn of some of its potential beauty. Master players like Warren Vache have focused their efforts on playing the cornet like a cornet, exploring its own potential for sound. And it can be stunning!
Do you absolutely need a cornet? Probably not. But, should you consider it, I would invite you to explore the horn as a unique instrument apart from your trumpet. You may find it offers you an entirely new voice with which you can express yourself.
If you are working as a professional trumpet player, there is almost no way to avoid needing a flugelhorn, unless you are specializing in a style of music that never calls for it. Almost every modern form of music that calls for brass will eventually require the trumpet performer to pick up his flugelhorn at some point.
While the act of playing the flugelhorn is basically the same as playing a trumpet, there is a far greater difference between these two horns than between the trumpet and cornet. You simply cannot approach the two horns the same way, although, sadly, many players do. The flugelhorn is a gorgeous sounding horn when played correctly and it is perfect for creating beautiful melodies. (I have written another article the explores the flugelhorn further).
Basically, you alone can decide when it's time to get additional horns. On one hand, you may not feel any need to add any horns beyond your sole trumpet—perhaps the opportunities aren't there yet for you. But, you may also find yourself creating more opportunities when you become known as a player who can offer it all. Whatever you decide, I hope you enjoy the process!