Why can't astronauts just take off from a runway instead of the rocket launch pad?
In theory they could. But no airplane invented yet can go even 1/5 as fast as orbital speed, and if you are going slower then you start falling down (or gliding down) as soon as your fuel runs out. A rocket has such powerful engines it can accelerate straight up, getting to space in under 2 minutes, where it then turns sideways and spends the next 8 minutes getting up to 17,500 MPH orbital speed. Such a rocket is more efficient in a vacuum and anyway no plane or rocket could go that fast in the air without burning up from friction with the air. Also, rocket engines need a ridiculous amount of fuel and burn it ridiculously fast, so rockets are really heavy with fuel at takeoff. And they are BIG, with most of the size being the fuel tank. Look how big that orange fuel tank is that the Shuttle has! No wheels or wings ever invented are strong enough to allow such a heavy vehicle to roll down a runway and fly like a plane.
What are the main uses for rockets?
Launching payloads into orbit, further launching things out into space and points beyond, they're used on US military transport aircraft (the aging ones) as JATO (Jet Assisted Take-Off - which is really Rocket, not Jet, but it was coined in the Jet Age, when everything was "jet") to get an aircraft into the sky carrying a heavier payload than it could normally lift, and of course, in all sorts of military applications on the battlefield, from anti-tank rockets to long-range area-denial rockets, to nuclear-tipped ICBMs and SLBMs. Air to air and surface to air missiles are rockets. Fireworks are not rockets - they're mortars, which are very different (of course, you can get those 2' long "Chinese Rockets" on July 4, but that's about it). The first supersonic aircraft were rocket-propelled, not jet propelled. Chuck Yeager's Bell X-1 was a two stage rocket. Rockets are used by hobbyists (from small to the 100,000 foot altitude kind), and I've even seen some people in... I think it was Colorado who study lightning by launching a rocket into a thunder cloud, with a copper wire trailing back to its launch pad, which directs a bolt of lightning into a predictable area and spot. Basically though, in the large vie, rockets are either something having to do with space, or something having to do with blowing something up. I've tried very hard, and I can't think of any common use outside of model rocketry that has anything peaceful to do with rockets.
Which space rocket can carry the largest payload into earth orbit? And how much does it cost to launch?
The Saturn V COULD lift a lot but it's no longer in service. For current vehicles, you'd need the Atlas V rocket - An Atlas V HLV DEC/5H2 - to lift 29,400Kg, which is the largest currently operating launch vehicle. However, in the first quarter of 2013, SpaceX's Falcon Heavy, will have its first test flight and will enter commercial service either that year or in early 2014. It will be capable of lifting 53,000Kg. The launch cost for an Atlas V isn't public domain but most informed speculation puts it around 124 to 187 million dollars per launch & the heavy-lift version would be closer to the upper figure rather than the lower. Falcon Heavy, on the other hand, would cost (and this IS public domain information - SpaceX is a remarkably open company in this regard) 80 million to 125 million dollars. So, today, it's the Atlas V heavy-lift variant lifting 29,400Kg at a cost of close to 190 million dollars. Coming very soon now, it's the Falcon Heavy lifting 53,000Kg at a figure of 80-125 million dollars. EDIT: @Martin, NASA is NOT building a 700 tonne payload rocket. Not even close. It IS building a 70-tonne-payload rocket (SLS) which will upgrade to 105-tonne-payload in 2021, upgrading further to 130-tonne-payload in 2032 but the cost is frankly ridiculous. 18 billion dollars to develop it from 2012 to December 2017, 3 billion dollars a year. It's expected to cost 41 billion dollars by 2025. The unit cost is expected to be 1.6 billion dollars which, when compared to the Atlas V & Falcon Heavy is absurd. In short, I fully expect SLS to be cancelled, next year if Romney wins, or in early 2017 when the next President moves into the White House.