Hi, First I have to warn you that making something like that using the vex robotics system will be very difficult. You have been warned.
Also, judging by the list they have given you, it seems they are recommending that you use only one motor to drive the slides, and chain them together. I would disagree with this. I would instead, buy two motors. This means you will be able to drive the linear slides separately in order to get more torque, and you wont have to buy the chain kit. (You can ask for clarification about this if you want it). Also, I am assuming you would control the extension of the arm with a button, but please correct me if you had something else in mind.
Anyway, I have put comments in bold next to each item on your list.
http://www.vexrobotics.com/products/accessories/power/276-1491.html Yes, this is a robot battery, and it will be needed to power your robot.
http://www.vexrobotics.com/products/accessories/power/276-1456.html This is another type of robot battery, however, since you already have one, it will NOT be necessary to get a battery of this type.
http://www.vexrobotics.com/products/accessories/logic/276-2194.html dont think i need this one. **This is the cortex microcontroller. It will allow you to program it in order to make motors go and receive inputs from sensors such as buttons. You will need a microcontroller, however for your application I would recommend the cheaper Pic v.5 Microcontroller, because it is cheaper and should do everything you want it to. http://www.vexrobotics.com/products/accessories/logic/276-2170.html **
http://www.vexrobotics.com/products/accessories/power/smart-charger.html This is used to charge the batteries. You will need one.
http://www.vexrobotics.com/products/accessories/motion/276-1667.html This is a motor, in order to drive the extensions. As I said earlier, I would buy TWO of these.
http://www.vexrobotics.com/products/accessories/motion/276-1096.html This is the linear slide kit. You will need this, as it will enable the extensions on the arm to slide outwards.
http://www.vexrobotics.com/products/accessories/motion/276-2166.html As I said before, I don’t think you will need this as I think you should drive each extension with a motor instead of using chain to drive both with one.
http://www.vexrobotics.com/products/accessories/motion/276-2184.html Yes, you will need this, as it has the rack gears for you to drive the sliders with.
http://www.vexrobotics.com/products/accessories/structure/276-2161.html This is a kit containing all of the metal, bolts, nuts etc. that will be required to build the arm. If you are very experienced in building metal things, this isn’t completely required, as you could just build the physical structure completely by yourself, however, this would make it a lot easier.
Additionally I also suggest you buy these:
http://www.vexrobotics.com/products/accessories/motion/275-1188.html These are gearboxes which will enable you to drive the linear slides using the motors, and the advanced gear kit.
http://www.vexrobotics.com/products/accessories/logic/276-2186.html This is a programming cable which will allow you to download a program onto your microcontroller. It connects to your computer. To use this you will also need to use the program EasyC or RobotC, but you can get 30 day trials of these for free of their websites (use google). If you need assistance for programming there are many experience programmers on these forums
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I hope this has been helpful to you, If you have any questions, don’t hesitate to ask.This is a Bumper switch. It returns a value to the micro controller when it is pressed. In your application, you could allow the expansion/contraction of the arm to be controlled by the press of this button, mounted somewhere on the arm.](“http://www.vexrobotics.com/products/accessories/sensors/276-2159.html”)