Riding experience of 21-speed vs. a 7-speed?
Do you use the front derailleur?  It seems many people in reasonably flat areas do not.  If not, you are riding a 7 speed.  Take a ride and notice what gears you are using, particularly on the steepest climb and the steepest descent.  The lowest gear will be with the smallest chainring in the front and the largest cog in the back.  The highest gear will be with the largest chainring in the front and the smallest cog in the rear.  Your gear ratios come from dividing the number of teeth on the chainring by the number on the cog.  Now you can compare the ratios available on the new bike to what you are using on the old one.  If you are having trouble up hill, do not accept a bike with any higher a low gear.  If you want to pedal down hill, do not accept a bike with any lower a high gear than you use now.

If you are using all the gears you have, a new seven speed will either have a much narrower range of gears or much wider spacing between the gears.  The spacing won't be a factor 3 wider because the gears overlap.  If the new one has the same range, try shifting two gears at a time on the old bike.  Is that too large an increment?  If so, you need more than seven gears.