Pre/Post Ride Bike checklist:

Before a ride & after a ride there are always a few things you should check over on your bike.

At a minimum you checklist should include checking:
~ Tire pressure
~ chain tension
~ quick releases (tightened)
~ lube chain
~ proper function of your shifters and brakes.

An expanded checklist might include:
~ Look for wearing and cracks in your cables & brake lines
~ Wheels straight?
~ Inspect the joints and welds of your frame for cracks and chipped paint
~ Tears in your tires (in pattern and sidewalls)
~ Overall tire wear
~ Inspect brake pads
~ Chain length
~ Inspect cassette and chainring teeth for excessive wear.
~ Nuts and bolts tight on: Cranks, pedals, handlebar, stem, headset, seatpost clamp.

Personally, I like to clean my bike after each ride, this usually means using "Honda Polish" (mc polish, similar to furniture polish for bikes) or good old mild soap & water. This really helps you look over your bike well. When drying off, look for paint chips, stress cracks, and cracks in the welds of your frame.

Tool To Carry With You On A Ride.

Basic Carry Along Tools : At minimum you should have some basic tools to fix those trail/roadside bike issues. Find what are called a Multi-tools which are "cycling specific" tools so they only give you the common tools needed to fix most minor emergency issues.
Theses tools usually contain:
~ 2, 4, 5, 6, & 8 mm Allen (Hex) wrenches
~ Both a flat & phillips drivers

Then more complex multi-tools may also carry:
~ Levers
~ Chain breaker (Tool used to remove/set pinned sections of a chain)
~ 10 & 12mm open wrenches
~ sharp blade(s)

The tools will range is price from $10-50. >

Fixing flats: Consider carrying a spare tube as well as a patch kit, Tire Levers (minimum 2 & steel cored are the best), & air pump. All of these tools can be carried in convenient little under the seat bag.