Quick tips

A team without an effective Engineer is at an extreme disadvantage, especially while playing Defense on Attack/Defend maps.

Always stay close to your buildings. Unguarded buildings are especially vulnerable to Spies, Demomen, and Soldiers.

The Teleporter can be your most important contribution to your team's success. If properly placed and defended, a Teleporter will allow your team to keep the pressure on the enemy team.

All buildings build faster during construction if you repeatedly strike them with your melee weapon.
You can help friendly Engineers by hitting their buildings with your melee weapon to speed up the building's construction or repair and upgrade the existing buildings.

You can construct a building in safety, then right-click to pick it up and haul it to the battle. It is also possible to carry buildings through Teleporters.
However, if you die while carrying a building, the building is destroyed.

Your Shotgun and Pistol are not useless in combat; use them to supplement your Sentry Gun in whittling away at the enemy's health.
The Shotgun can kill most enemies in 2-3 hits, and a well-aimed Pistol can kill most classes with a full clip.

Avoid standing in front of your Sentry Gun. This reduces the chances of your sentry shooting you, as well as lowers the chance of being killed, as the Sentry can intercept enemy fire.

When playing Arena, make sure you have Dispensers to help out teammates with ammo and health; there are no health packs in Arena mode.

Try to set your Sentry Gun at chokepoints where enemies frequently prowl, thereby maximizing its effectiveness.
This is of huge importance when playing on Defense.

Spies can sap your sentry faster than you can remove it. Kill the Spy first, then get rid of the Sapper.

Try to construct your buildings in a location where not only can you support your team, but where your team can support you. While you and your Sentry might be able to hold ground on your own for a short time, multiple enemies can quickly overwhelm you. Be sure to ask your teammates for help as often as possible while you help your team at the same time via your buildings.

If possible, try to build your Teleporter and Dispenser first. These buildings allow both you and your team to get to the frontlines faster and keep you all in the fight due to the constant flow of health and ammo.