How to decide on a rate of climb or descent if no specific vertical speed is required?
In general terms, the higher you are the better your fuel efficiency, so you will want to climb quickly and then stay as high as you can until you need to start your descent in order to land.  For jets, a common way to approximate your top of descent is by using the "3:1 rule" where you take your desired altitude loss in thousands of feet and multiply it by three.  For instance if you need to lose 35,000 ft. you would start down 105NM ($\frac{35000ft \times 3NM}{1000ft}$) prior to your destination, staying at altitude until then in order to save fuel.

As far as your rate, the first place I would check would be the AFM of the airplane that you are flying.  Smaller airplanes will have Vy (best rate of climb speed) that may be used for this purpose unless another speed is specified (to help with cooling, etc).  Larger aircraft often have specific climb and descent schedules (for example, climb at 250 KIAS until 10,000 ft then 300 KIAS until you transition to Mach 0.76) which the manufacturer recommends and provides performance information for.  Fly these speeds and take what vertical speed that you get.  Usually, flying by the book is the place to start unless you have a specific reason to do otherwise.

The AIM 4-4-10 says:


  d. When ATC has not used the term “AT PILOT'S DISCRETION” nor imposed any climb or descent restrictions, pilots should initiate climb or descent promptly on acknowledgement of the clearance. Descend or climb at an optimum rate consistent with the operating characteristics of the aircraft to 1,000 feet above or below the assigned altitude, and then attempt to descend or climb at a rate of between 500 and 1,500 fpm until the assigned altitude is reached. If at anytime the pilot is unable to climb or descend at a rate of at least 500 feet a minute, advise ATC. If it is necessary to level off at an intermediate altitude during climb or descent, advise ATC, except when leveling off at 10,000 feet MSL on descent, or 2,500 feet above airport elevation (prior to entering a Class C or Class D surface area), when required for speed reduction.
  
  NOTE- Leveling off at 10,000 feet MSL on descent or 2,500 feet above airport elevation (prior to entering a Class C or Class D surface area) to comply with 14 CFR Section 91.117 airspeed restrictions is commonplace. Controllers anticipate this action and plan accordingly. Leveling off at any other time on climb or descent may seriously affect air traffic handling by ATC. Consequently, it is imperative that pilots make every effort to fulfill the above expected actions to aid ATC in safely handling and expediting traffic.