Mouthpieces
Trumpet players should start on a Bach 7C. With Bach mouthpieces, as the number decreases, the size of the mouthpiece increases, specifically the cup diameter. The letter refers to the cup depth. As you progress through the alphabet, the cup gets shallower.
When the student can play from low C to top-of-the-staff G with good tone and is practicing 30 minutes/day, the student is ready to move to a 5C.
When the student can play from low F# to high C and is practicing 45 minutes/day, the student is ready to move to a 3C. The student can move from a 7C straight to a 3C if they meet these criteria.
Once on the 3C, the student can remain here throughout high school. Any subsequent changes should be made with a private instructor.
(Progression concept borrowed from Michael Huff, https://www.troy.edu/academics/colleges-schools/college-communication-fine-arts/departments/john-m-long-school-music/faculty-staff.html).
The student may regress at first when using a larger mouthpiece regarding range and endurance. This is normal. Be patient. The student should not force progress. With patient, gradual practice, the student should eventually sound even stronger than before.
Do not move students to a particular mouthpiece size by age. Move them by the criteria above. If a trumpeter does not advance beyond a 7C during high school, so be it.
Do not succumb to the temptation of switching your trumpet players to smaller mouthpieces in order to gain more range. For example, the Schilke 14A4A should be avoided. High school and younger trumpeters should be playing only one mouthpiece as described above unless the trumpeter is very strong, and the private instructor feels a different mouthpiece might be more suitable for jazz or marching band. High school and younger trumpeters should not be experimenting with mouthpieces. Mouthpieces do not solve playing problems, practice does!
I do not recommend purchasing used mouthpieces unless you have evaluated them in person, and they are in like-new condition. Mouthpieces should not be worn, scratched, dinged, or dented.
Trumpets
Recommended horns for beginning students:
Yamaha student model, examples include YTR-2330 or 200AD, new around $1600 Or
Bach student models, example includes TR600, new around $1000
It is a good idea to rent a horn at first, but once the student seems like they will stick with the instrument for a while, I recommend purchasing a used beginner horn. It will be more economical than renting. Good used trumpets of these and various other brands can be found on eBay and www.trumpetherald.com. If you purchase a used horn, make sure there is a trial period during which you can return the horn with a full refund. For beginner horns, the make and model of the horn is less important. Most important is its condition: Are there any major dents or a lot of lacquer wear? Do the valves move smoothly and easily? Do the slides move (this is sometimes an easy fix)? It is best to have a trumpet player you trust evaluate the horn.
Professional models:
Yamaha Xeno 8335, new around $3,300
Or
Bach Stradivarius 18037, new around $3,000
I do not see a purpose to an intermediate level horn. Once the student has moved to a 3C using the criteria above, they are ready for a professional model horn.