What is the normal "feet per minute" an airplane travels at to get to it's cruising altitude?

Let's say I'm flying a McDonnell Douglas/Boeing MD80 and I'm climbing to my cruising altitude (20,00ft [FL200]). How fast should I climb to it? Does this depend on the weight of the plane (payload, fuel)?

You'd have to be more specific. There are many different makes and models of airplanes, just like cars. Only in the sky there is no speed limit, just whatever the cruising speed of the airplane is.

Every aircraft from rotor to fixed has a manual which gives the best angle & the best rate of assent.Climb rate IE: FT per min.is of course is dependent on the weight, power setting, pressure altitude, angle of attack and if course the Pilot. A normal rate of assent from an airport would be the best "rate" to get the aircraft up and away from traffic
Were changes in route would normally be the best "angle" as not to empeed forward Air speed.

Rate of climb will depend on weight of the plane, power setting and airspeed. In an MD80 or any other passenger jet, the climb rate will be very high at the start of the climb, but will decrease as you get higher. I'm not familiar with the MD80 enough to tell you exactly what you should climb at. If you are playing a simulation game a nose up pitch of about 8-10 degrees, at climb power setting will get you close to the proper speed and climb rate for almost any airplane. You will probably average about 1500-2000 feet per minute in the MD80.

Rate of climb will vary considerably depending on weight, density altitude, air traffic, etc.

An MD-80's initial climb will be around 3,000 feet per minute and then taper off to around 1,500 - 2,00 FPM past about 5,000 msl.

Once you're above about FL300, you're gonna get only about 1,000 fpm.