Must haves

Fuses: once you get on your own, buy a box of assorted fuses. Learn where they go. There are fuses under the hood and fuses in lines above the batteries.

A tool box with a decent knife: as you go along you'll add stuff like electrical tape and electrical heat shrink wrap.

Learn how to adjust trailer brakes and carry the tools necessary. I have to use a mini pitch fork thing to back mine off.

I also keep a 4 lb small sledge hammer with me.

I'm sure there's some stuff I've left off but someone will fill it in

 

Earplugs are nice when you need them.

 

As far as a student I couldn't tell you... Never had to ride with a trainer. But for solo...

Tums
Toilet paper
Flashlight
Another flashlight
Cigarette lighters
Ink pens
Tablet... The paper kind.
Motor Carriers Atlas, in physical form not digital.
Extra fuel, oil, and air filters
Extra oil
Washer fluid
Duct tape
Baleing wire
Super glue
Shovel and shop broom
Enough winter fuel treatment to treat 2000 gallons (if anywhere even remotely cool)
Tire pressure gauge
60 foot air hose and tire chuck.
Extra glad hand gaskets
Extra trailer pigtail
Tire chains
First aid kit
20+ foot heavy duty jumper cables
Tools
Extra lights
Extra windshield wipers
Extra compression rings and bullets for air lines from tractor to trailer.
Wheel chocks
Cash
Food, water and clothing to survive inside that truck for 3 to 5 days in -20 F degree temps.
Extra batteries for anything else you have that needs them.

I'm sure I'm missing some stuff but that is most every thing that I ALWAYS have (with very few exceptions)

 

First Aid Kit - This is the stuff my husband has in his at all times in the truck. I carry a duplicate in my trailer when I'm hauling horses.

Multiple size Band-Aids
Gauze
Paper towels
Antibiotic ointment
Burn gel
Benadryl cream
Liquid Benadryl (for severe, sudden allergic reactions)
Wound wash with lidocaine
Good tweezers
Rolls of Vet Wrap (self adhering bandages - no need for tape)
Finger splint kit
Sealed wet wipes
Aspirin, Tylenol and ibuprofen (each does something slightly different)
Kwikstop Blood Powder (stops/slows bleeding)
Tourniquet - know how and when to use this
Suture kit - only if you have training for this
Small shock blanket
Ankle and wrist braces
Eye drops/ eye wash

At some point we've used everything on this list over the years. We carry a second first aid specifically for any dog(s) traveling in either his truck or with me as well.

 

might I suggest 2 "how to win friends and influence people " by Dale Carnegie and a " Dummies guide for tractor trailer driving"

 

One can get nice Bolle sunglasses at the Shepherd's Hills outlet center in Lebanon, Missouri, off I-44. That's the same place that has the Case knife outlet store in it. I stop in there about every 6 months and get a new pair. They have Bolle wrap arounds that are polarized and UVA-UVB 400 for about 30 to 40, normally 130-160.