How to reduce the bitterness of tonic water in a non-alcoholic citrus drink?
EDIT: My original version of this answer came from my incomplete recollection of a chapter in Kevin Liu's Craft Cocktails at Home on flavor balancing.  Now that I have the book in front of me again, I'm adding more relevant detail and revising the parts I got wrong.  In all fairness, salient points are already covered in other answers, but I think the science here is neat.

Bitter flavors in general are fairly difficult to mask.  This is possibly an evolutionary adaptation - bitterness can be an indication of toxicity in wild plants, so a sensitivity to bitter compounds in very minute quantities may have helped our ancestors avoid being poisoned.  In order to produce flavors of roughly equal intensity (using a perceptual measurement technique known as the Labeled Magnitude Scale), it takes about 3000 times the concentration of sucrose (sweet) to equal one part of quinine sulfate, the principal bittering agent in tonic water.  (It takes about 1500x relative concentration for sodium chloride and 50x for citric acid).

Relative concentrations aside, you get some interesting effects when these flavors begin to interact.  (Liu reproduces a number of figures from a fascinating paper published by Green et al. at this point.)  Perceptually, sweetness is also very difficult to suppress (another likely biological preference, since sweetness often indicates a concentrated source of energy in the form of sugar) and it will tend to dampen the perception of other flavors.  This is one of the reasons why commercial tonic water typically includes a pretty hefty amount of sugar or corn syrup (somewhere around 30 grams per 12 oz).  Adding sourness to the mix will suppress bitterness further; fortunately you're already doing this by using citrus.  Finally, salt will substantially reduce the perception of bitterness, while remaining barely detectable in the final mixture.  Salt also has the unique effect of "leveling out" sourness and bitterness when added to a solution that contains both (compare the rightmost two groupings from this figure) and as a result, Liu concludes that "Salt = magical fairy dust".  Yes, that's a direct quote from the book.

The lesson to draw from all of this makes intuitive sense: since bitterness is so difficult to cover up, you want a lot of other flavors in combination.  Adding sugar for sweetness will help, and including acid from citrus will too; it sounds like you've discovered this on your own.  The counter-intuitive part is that adding salt will further suppress bitterness and mellow out the acidity as well.

I would add a couple other observations from personal experience.  Because bitterness is detectable in low concentrations, you definitely don't want to add any more.  Steer clear of citrus that has a significant bitter component (like lime or grapefruit) and be careful to avoid introducing pith (the bitter white part of the peel).  If you're adding sugar, you can be much more liberal about the amount of salt you add; sweetness has a strongly suppressing effect on saltiness.  You could probably get away with a teaspoon or so in a gallon of tonic water if you're adding a couple tablespoons of sugar too.  As others have mentioned, this will all be much easier to combine if you make solutions of the sugar and salt (for simple syrup and, ah, salt water respectively) and then combine those into your tonic instead of just adding the dry forms. 

Finally, don't give up on the tonic too soon.  It's a bit of an acquired taste, but I find that slight bitterness can lend some depth to mixes, and some guests really enjoy that; there's a reason why the gin and tonic is so popular.  If the quinine is really too much for you even after extensive modification, try diluting with additional seltzer or club soda.  Don't toss it out; instead, try using it in small dashes in other drinks as you experiment.  Trying new combinations is the absolute best way to learn what you like.