Objects falling through the atmosphere accelerate from rest to reach a so-called terminal velocity. (For an adult human, this is about 120mph). Parachutes make it possible for humans to fall with a terminal velocity much less than this- 10-20mph, say.
Once the terminal velocity is reached, then this remains steady, and a properly designed parachute (designed, that is, for the load it's carrying) reaches terminal velocity fairly quickly. To answer your afterthought question then, yes, it will take twice as long to travel twice as far.
The weight of the whole package (parachute plus load) as well as its shape will determine how fast the whole will fall.
The bigger the parachute for a given load, the slower it will fall. The bigger the load for a given parachute, the faster it will fall.
So for a given load and parachute shape/size, a light parachute will fall slower than a heavy one (silk is reckoned to be a suitable material, along with recent man-made fabrics. Strength and lightness are important).
A sensible start could be made by cutting a circle out of an easily-obtained sheet of thin plastic/polythene, and making a shallow cone from it (as already suggested), and tying three or four nylon threads to it around the rim, to suspend a light plastic cup, acting as a load carrier. The sheet should be as large as you can comfortably handle, and the material as thin as you can find.
You should then, using as many different weights as you can, time the passage of the parachute between two fixed vertical markers (again, as suggested, down a stairwell). The time taken will very much depend on how big your parachute is, and how heavy it is, as well as the loads being dropped. You will need to devise a way of making the actual timings (your decision- but remember, someone's got to let go of the 'chute, and someone else needs to record the times when the 'chute passes the top and bottom markers).
You could repeat the experiments using different sizes of parachutes, but with the same load. This will be much harder to quantify, so I should stick with a single parachute and different weights!
There are a few sources of error- you should be able to think what these could be {clues- how could you ensure that the 'chute had reached its (steady) terminal velocity? How's your timing accuracy- have you got a stop-watch? How about measuring the distance between the markers? Is it draughty in this stair well (or wherever)? Will the atmospheric pressure make a difference- if so, how much? How about the air temperature and humidity?}
This is a good project to exercise your experimental skills and your scientific thinking, too!
Good luck!