RyRNick,
Reverse thrust is simply a way of using aerodynamics to help slow the airplane, rather than so much brake. Airplanes have a lot of weight, and very small brakes. Any deceleration created by reverse thrust from the engines is less energy the brakes must absorb, and therefore lower brake wear and temperatures. Often the biggest limiting factor on how fast an airplane can be turned around at the gate is brake temperatures.
Reverse thrust is very simple. Just like your car engine, the jet engine requires a lot more power to do the work than the work it actually does. What this means is that the net thrust produced by the engine is less than the total power it produces; a lot of the power the engine produces is taken up by moving internal components and by keeping itself running. This includes overcoming internal drag. Your car is the same way; the engine produces a lot more power than what it actually puts out; most of it is required to overcome the internal friction and mechanical resistance of the engine itself, from turning oil pumps to other accessories, to overcoming spring tension in valve trains, to the friction in bearings, etc. The jet engine is the same way; it has accessories, and uses much of it's own thrust to turn it's compressor.
Take away the useful thrust being put out by the jet engine, and what you have left is a lot of drag. When reverse thrust is applied, the actual redirection of the exhuast gasses isn't what's producing the deceleration effect, though it helps contribute. It's the drag that's left over when the thrust portion is taken away. Blocker doors of varying types, or aerodynamic "cascades" alter the direction of the airflow being put out by the engine; the thrust or exhaust gasses are often directed sideways or sometimes slightly forward, instead of out the back of the engine, by blocker doors. Taking away the thrust leaves only the drag...this means the brakes need to do less work to produce the same slowing of the airplane on landing.
The exhaust gasses which went beneath the wing and aft during normal operation, are suddenly made to sound much louder as they are directed by the cascades or blocker doors or vanes in an outward direction. In a sense, it's as though you were suddenly placed beyind the engine; you're now in line with the exhaust gasses instead of having them diverted below and behind you in normal flight. It seems louder. The engine may shake and vibrate somewhat on it's pylon. It's designed to do that, as is the pylon mount.
All landings are calculated without the help of reverse thrust. Reverse thrust is therefor a bonus in helping the airplane slow down, achieving less wear and tear on the brakes, and quicker ground turns when going back out for another leg with passengers or freight.
As others indicated, the vibration you feel as the landing gear is coming up after takeoff, is simply wheel vibration due to balance. Generally the main gear will incorporate a feature in the brakes to slow the wheel and stop it from spinning as the gear retracts. Until that time, however, any imbalance in the tire will result in a shaking or vibration which will decrease in frequency as the wheel slows. The imbalance is created through landings when the tire is spun up from a complete standstill to the landing speed of the aircraft. You may have noticed a puff of smoke when an aircraft lands; that smoke is loss of some tire material as the wheel touches down and is quickly spun up. This loss of material naturally causes a slight imbalance, and this in turn causes the vibration you feel.
Wheel assemblies are typically balanced when being built up...much in the same way that a care tire is balanced before being put on your car.
When the airplane comes off the ground, the vibration will be felt until the tire assembly stops rotating. Often the main gear will stop rotating fairly quickly, due to some back pressure which is induced in the brake lines during the retraction process. The nosegear, however, may continue to spin for a short while in some airplanes, while others incorporate a mechanical snubber in the gear well to bring it to a stop.