What kind of horn do you recommend?

This is one of the hardest questions to answer.  Personally, I prefer Conn horns.  However, just because I, or anyone else, recommend a certain horn, doesn't mean that horn is best for you.  I know someone who sounds great on a Holton horn.  However, when they pick up a Conn, they don't sound quite as good.  

To make a decision between horns, you must try several different types of horns out.  Listen to how you sound with each one.  If you like how you sound on the horn, then that is probably the one for you.

What mouthpiece do you recommend?

Again, a tough question to answer.  I have three favorites.  My favorite and the one I currently use is a Lawson mouthpiece.  The great thing about this mouthpiece is that you can get many different cup sizes and shapes as well as rim sizes and shapes.  It is a very customizable mouthpiece, so you can get a mouthpiece that fits you well.

Two mass-produced mouthpieces that I recommend are the Giardinelli C8 and the Schilke 31 or 31B.  I feel both are a good "middle-of-the-road" mouthpiece and do generally well in all the registers.

What kind of case/gig bag do you recommend?

This is a little less subjective than the question above.  I use and recommend a Marcus Bonna gig bag.  This case is compact, durable, and very protective.  The great thing about this case is that it has both a layer of padding and a fiberglass shell to protect the instrument from knocks and bumps.

Another great gig bag is made by Reunion Blues.  These cases are inexpensive and extremely light.  Also, they have a really thick layer of padding that protects from most bumps.  The downside to this case, that I have found, is that more severe bumps will put a ding in the horn.  But, I do highly recommend this as an inexpensive alternative to the Marcus Bonna.