For those in need of services:

Click the "Client Portal" link at the top and login

Click "New Service Request" and fill out the form. The more details, the better.

Wait for a Contractor to accept your request

Chat with your Contractor and be ready to accept their friend request in-game

(If you are down and bleeding out, make sure you aren't in third person to accept the friend request)

For those providing services:

Click the "Contractor Portal" link at the top and login

Set your Star Citizen Username

Select which ships you are ready to fly and provide services with

Click the "Contract Feed" button to be brought to a list of available contracts (the page will auto-update, no need to refresh)

Accept a contract, chat with your client and provide them services

(Add them as a friend to get into their server)