The main power delivering part of any [non-supersonic flying!] turbine engine these days is the cold section of the engine. Reason for this is that hot air holds a lot of energy. Blowing this hot air into the atmosphere through any form of exhaust ruins the efficiency of the engine since a lot of energy is being lost by blowing it over board so to speak. Mind you, [hot] gas-energy basically is present in three forms: temperature, pressure, and velocity.
The hot section of a modern turbine engine produces a lot of hot air, at high pressure and lots of speed. So to make maximum use of this energy, additional turbine stages are being put in place behind what we call the core engine [basic engine - gas generator].
These turbines convert the hot gas energy into mechanical energy [energy which would otherwise blow off into atmosphere, thereby losing a lot of potential energy/efficiency]. That mechanical energy is now being used to drive a large fan [turbofan], or a propeller [turbo prop], or a shaft which can drive anything from a helicopter rotor to a natural gas pumps or electricity generators or even ships [turbo shaft].
Utilizing this energy to drive a fan increases efficiency dramatically, since the lost energy from a fan is much smaller than from the hot end of the turbine. The airflow which the fan is blowing into atmosphere is much cooler and slower, thereby reducing lost energy. The down turn off course is weight. One would like to make the fan as large as possible; a larger fan turns slower and thus less energy loss. But the larger fan is also very much more heavier [which you don't realy want in an airplane]. Losing the shroud would help very much in terms of weight --> propeller/propfan. However due to aerodynamic effects, the shrouded prop [better known as fan] is better suited for high speed [Mach 0.85] where as the prop or propfan is better suited at M0.5 - M0.7.
Now coming to your question, the turning speed of a fan very much depends on the size of the engine. I'm not into big engines, but I believe the fan of a GE CF6-80 on the 744 will turn at approx 3400 rpm [=3400/60 = 56 cycles per second].
On a modern turbo fan, upto 80% of the thrust is being produced by the fan, the balance coming from the hot end.
On a small engine like the 2100shp PW120 series [to be found on Dash 8 and ATR42] the power generating turbine rotates at approx. 20000 rpm. Mind you, this rotational speed is being reduced by a reduction gearbox [RGB] to approx. 1050-1200 rpm, which is the optimum prop speed. The high pressure turbine [which drives the high pressure compressor] rotates at 33000 rpm.
Theoretically, this same hot section and compressor [which are referred to as Gas Generator] can be used to drive a small fan. There are engines where basically the same Gas Generator [aka core engine] is used to drive a prop in one engine model, and a small fan in a different engine model from the same engine family.
Scootertrash is right in stating that 75% of the energy produced by a turbine engine is used to drive the engine itself [i.e. the compressors]. however this might be very misleading without any additional comments. One should not forget that this 75% energy is being returned into the gas stream by the compressors. The only thing that a compressor does is converting mechanical energy [i.e. rotating spool] into gas energy [minus a couple of % efficiency loss].
So lets say that you burn 100 units of energy through fuel. 75 units of this is required to drive the compressor. 75 units are being returned into the gas path, These 75 units enters the engine, and another 100 units of energy are added by burning fuel. So I now have 175 units of energy in my hot section, again 75 units required to drive the compressor. I now have 225 units of energy in the gas path et. etc. etc. End game: all the energy required to drive the compressor is being returned into the gas stream and is NOT lost, meaning that all 100 units of energy inducted through fuel burn [minus a couple of percent due to turbine/compressor efficiency loss] are still available for net power delivery.
Total thermal efficiency of a typical turbofan is around 40-45%, meaning that 55-60% of fuel energy is lost. Compare this to a typical automobile engine which has a thermal efficiency of only 20-25% . . . ! Yeah, that's right folks, 75% of the thermal energy in your auto gas tank is directely converted in heat, warming our global atmosphere, and only 25% is used to move your car and yourself [which off course is also converted in global heat due to friction].
Concluding: on modern subsonic engines, the core engine is working to drive the fan or prop or whatever. Approx 15-25% of the thrust is being produced by the hot section. The main power delivering source is the cold part of the engine.
PS. On a Fokker 50 [which has two 2500shp PW125B turboprops], the hot gas stream is angled 15 degrees or so downward, generating upto 5% of the aircraft lift requirements . . . reducing wing lifting requirement and thus drag by 5%!
Immigration officer: "What's the purpose of your visit to the USA?" Spotter: "Shooting airliners with my Canon!"