I know what you mean by 'soda', Anna - it's what we (ie in the UK/Ireland) know as bicarbonate of soda.
There's also a drink called soda water, which is carbonated water, but not quite the same as Perrier (though I couldn't tell you what the difference is!
).
I grew up in Dublin calling soft drinks 'minerals' - it was what everyone around called them. I think that's an Irish thing. Then they became 'fizzy drinks'. And then, when I moved to the UK, 'soft drinks'. In Canada, or at least in Ontario, 'pop' seems to be the most common term - but I can't quite bring myself to use that, because my mental image from the word is of a minor explosion!
In English, you wouldn't ask for 'a refreshment', however; it's either 'refreshment' or 'refreshments', without an indefinite article. But refreshments can mean food and drink, so if you asked for that in a restaurant it wouldn't be specific enough.
Better just to ask for a drink - as in "What drinks do you have?"
That's mostly what I do these days, in order to avoid the thorny soda/pop/whateverelseitmightbecalled question!
Wendy
