Whole articles have been written about the use of ice in drinks, and with good reason (Esquire magazine ran a very good article on the subject, which is well worth reading).
If you think about the ingredients by volume, then in a mixed drink, the ice is going to be the single largest item by a long way. Which does mean that the quality of ice is going to have a significant influence on the quality of your drink. At the very least, this means choosing a source of ice rather than that single ice tray you have had at the back of the freezer, under the bag of peas, for the last year or so. Ice goes stale after prolonged storage, so fresh ice is an absolute must.
Secondly, tap water doesn’t make brilliant ice, it must be said, depending on where you live. Some people can detect a chlorine taint, or a distinct flavour, so the simplest solution here is to run your water through a Brita water filter first if you are planning to make your own – or, to get rid of the chlorine hint, let the water stand in a jug for a while before freezing.
The next thing to realise when you start mixing drinks is the sheer volume of ice you will need: a single shaken cocktail will need enough ice to fill the jar section of your Boston shaker, which is about the quantity a single freezer ice tray will produce at a time.
The simple solution to all of this is to plan ahead and buy your ice in bulk from a supermarket. Commercially-made ice is slow-frozen (so the cubes are attractively clear), and made from filtered or natural spring water – so has no discernible flavour. My local Tesco has filtered-water ice cubes in bags for around £2, and I find three bags are more than enough (usually) for a cocktail evening with friends.
The last solution is to buy a home ice-maker. I found a second-hand one on eBay for £20 which has proved to be very effective – from filling to first ice takes around 25 minutes, and I run it for a few hours, bag the ice & store it in my home freezer for use over the following few weeks. I also filter the water through a Brita jug first. The only downside is that the quick-freezing process used by the ice machine produces characteristically ‘cloudy’ cubes. I don’t find this a problem for my own drinks, but if I am friends over, I will buy the clear bagged ice, as above.