A properly-fitted, well-maintained bike shop (not department store) quality mountain bike.
Your local bike shop can advise on an appropriate bike for your rider.
A properly-fitted bike helmet in good condition.
Closed-toe footwear. Bike-specific shoes are recommended for comfort and durability.
Water bottle and frame-mounted bottle cage, or a hydration pack.
Handlebar-mounted water bottles are not permitted for safety.
Spare inner tube that fits your wheels and tires.
Bike-specific multi-tool for adjustments and repairs.
A chain-breaker tool is optional
Pump or CO_2 inflator with cartridges
Tire levers
Spare chain master link that fits your specific chain for repairs in the field. Your local bike shop can help determine which one you need.
Front and rear bike lights.
Handle bar end plugs (this is a safety measure).
Necessary medications (albuterol inhalers, epi pens, glucagon, etc.)
Team jersey (available for order at the beginning of the season)
Eye protection
A lightweight, wind-/water-resistant (but breathable) cycling shell for inclement weather
Bike gloves
Padded cycling shorts
Non-cotton athletic clothing
Frame bag, under-seat bag, or (hydration) backpack to carry riding essentials (tools, etc.)
Sufficient snacks for the planned ride (200 - 400 calories of simple carbohydrates per hour of planned riding)
Spare derailleur hanger that fits your bike
Kick stands
Bike Locks
Jeans
Bar ends
Anything that you do not want dirty
How to Carry Your Gear
Hydration backpack
This option allows the most flexibility to carry basic tools, snack, and a layer if weather is an issue.
Pros: flexibility, large capacity
Cons: can be heavy, not pleasant to wear when it's hot outside.
Frame bags
This option can be a good alternative, depending on the particular bike frame design. Some frame bags are designed to ride on the top tube and provide quick access to snacks.
Pros: weight is off of the rider
Cons: fit, vibration and clearance for riding can be tricky to work out. Also, they will tend to dull/scratch glossy frame finishes, due to grit and vibration.
Under-saddle bags
This option is very convenient for the minimalist. They usually offer enough space for basic tools and a tube, and of course can be left on the bike, which means it's harder to forget basic ride gear. Great as an option in conjunction with jersey pockets (see below).
Pros: easy, secure way to carry gear.
Cons: will get covered with mud, fit can be challenging with a dropper post (cannot clamp to post; tire clearance with full suspension bikes), limited space, interference with tail lights.
Cycling jersey with 3 back pockets
It's amazing how much can be carried comfortably in three pockets on the back of bike jerseys: light rain shell, snacks, light tools. Combined with a small saddle bag, this is a winner.
Pros: very accessible while riding
Cons: whatever you carry will get wet with sweat/rain/mud, etc.; does not work with a backpack
Small hip/fanny pack
The 90's called and left a voicemail... and this option offers many sizes/features, including hydration bladder options.
Pros: more comfortable in the heat, potential for high capacity, flexibility
Cons: do not carry well with heavy weight (water), waist belts can be uncomfortable while riding, long hydration bladder hoses can be difficult to manage.