After the intro and character creation, your character is regaining conscience following the explosion. A small robot, named Claptrap, wanders about the glacial area, burying the bodies of dead Vault Hunters. It suddenly notices you are alive. It is very happy to have found out you are not dead, and plans on using you to help it get off the island. Claptrap has funny dialogs, so listen to what he is saying. He gives you an ECHO Communicator, and this initializes your HUD (User Interface). Note the minimap on the top right corner, the health display on the bottom left corner, and the experience points display and level progress in the center bottom of the screen. You start at Lv 01, and as you kill enemies and progress on missions, you gain experience points and level up, making your character more powerful.
Claptrap starts to move away. Follow him. A mysterious woman, your guardian angel, contacts you through your ECHO Communicator. She tells you that you are alive for a reason and that she is here to help. She will explain everything soon.
For now, don't worry about your low health, which is flashing red. You will be picking up some health packs soon.
Keep following Claptrap. You reach a door, it scans Claptrap's Eye. Claptrap explains that it is to protect from Bullymongs, who would rip your eyes out at the first opportunity. Follow Claptrap into Claptrap's Place. Claptrap tells you that the creatures around here are dangerous, but none more than this Bullymong named Knuckle Dragger - killed everyone he knew.
Claptrap mentions he keeps a pistol in the cabinet, for emergencies, but that you should be pretty safe in here. Soon you here a roar, and Knuckle Dragger enters from the completely open roof above. He then proceeds to rip Claptrap's eye out. This begins your first mission in the game, "My First Gun". This is a STORY MISSION, meaning it is mandatory you complete it. There are also other missions that are side missions, and those are considered optional.
Notice also the diamond icon that appears on your mini-map. This is an objective waypoint, and tells you the general direction of where you need to go. When you are close to the destination, you will actually see the diamond in front of you. Under the minimap, it tells you the mission name, "My First Gun", and underneath that you should see your objectives. Your current objective is to open the cabinet. The cabinet is a crate-like object near the wall of the main room. Use the diamond waypoint to locate it. If you are still having problems finding the gun's location, see the screenshot below:
NPCs will sometimes give you hints on what you have to do if you appear stuck. So if you can't find the gun, Claptrap will keep calling "The gun... The gun in the cabinet..."
Open the cabinet. This first mission is an easy introductory mission, so as soon as you open the cabinet, the mission will advance to the "turn in" stage. When missions are completed, you turn them in to certain NPCs or certain locations. In this case, you just need to interact with the cabinet to complete the mission. You get your first gun, a "Basic Repeater". Note its stats. It has a damage of 11, which is how much damage it does per shot. The higher the better. Accuracy increases your chance to hit the enemy, while fire rate lets you shoot faster. Reload speed determines how fast you can re-load your gun, and magazine size is the maximum number of bullets a magazine can hold at once. Other attributes shown are the manufacturer (in this case, "DAHL"), the level requirement (none for this gun) and other bonuses, for example "Burst fire while zoomed".
As mentioned, you usually want to use the weapon with the highest stats. Though there will usually be a tradeoff, for example a weapon with a very high fire rate could cause less damage. Shotguns, for example, cause a lot of damage but have a slow fire rate. This is your first weapon, but you will find many many more.
The guardian angel who contacted you earlier contacts you again. She says that once upon a time, four Vault Hunters changed Pandora forever (This refers to the original Borderlands). But their time has passed. Thanks to Handsome Jack, Pandora needs a new hero, and that hero is you.
This completes your first mission, "My First Gun", and begins the next mission, "Blindsided". If it is your first playthrough, you also get the achievement "First One's Free", for completing the mission.