What do I need to do the following things with an arduino?
Regarding the first part of your question:  Are you using DC or AC? If it's AC, things are potentially a modest amount more complicated, and I have little experience in that world so I'll defer from commenting much further, on any AC specifics, other than mentioning Triacs

That said, in principle, there are a few ways to control a higher-current / higher-voltage load from an Arduino.  

Note that in all the scenarios I'm describing here, I'm talking about using an independent power supply to power the load. 


use a "power transistor."  You'd probably want a MOSFET or an IGBT, rated for the voltage / current you're trying to control.  For MOSFETs, you'll probably want to look for a "logic level" MOSFET where the gate can be driven fully on by the 5 volts from an Arduino digital pin.  If you use an IGBT or an non-logic-level MOSFET you'll need additional circuitry to drive 10-12 volts (usually) to the gate.  They make "MOSFET Driver" ICs that are purpose built for this, but you can also roll your own driver (of sorts) a couple of different ways.  I've gotten good results using a quad voltage comparator IC.
Use a (mechanical) relay.  You'll have to poke around the parts catalogs to find a relay that can be controlled by an Arduino, or - again - you'll need some additional circuitry between the Arduino and the relay.
Use a solid-state relay (SSR). 
If you're doing AC, you'll probably want a Triac.  


Some other considerations:  You might want an optoisolator between your arduino and ANY of these devices.  It's not strictly required, but it will help protect your Arduino from damage.  

IF your load circuit is DC, you can (usually) just join the ground of the power supply that's driving your load and the ground on your Arduino.  This makes wiring things up a bit simpler, but there is some small risk associated with ground loops. This is not a concern if you use an optoisolator.

If you're doing AC,  you absolutely can't just connect the two circuits together.

If your load is inductive (like, say, a motor) there are additional concerns regarding "inductive kickback."  Read up on that before you start switching motors on and off.  

As for measuring.. are you trying to measure voltage or current (amperage) or both?  For current, you could possibly use a hall effect sensor, or an inductive current sensor.  I've never tried measuring voltage like that, but there's a long thread on the old Arduino forum talking about this stuff:

http://www.arduino.cc/cgi-bin/yabb2/YaBB.pl?num=1205745528