Regarding drinks with Chinese food, I can only give my experience (after 18 years in central and southern China.)
As usual, it is different in restaurants and home situations.
In smaller, informal restaurants you will usually be given a cup of week green tea as soon as you sit down to be consumed as you peruse the menu. In the cheapest places you will be given a glass of hot water (Many Chinese believe drinking cold water is dangerous.) This is usually refillable for free, but few people do. Most people drink beer or local varieties of Lilt or Tango type soft carbonated drinks. The restaurants seldom sell the carbonated drinks but your are free to bring your own. They will sell beer, but often it is warm.
In more upmarket restaurants and banquet venues, a better tea is served at the beginning and perhaps also at the end to clear the palate before you go home. It would, in my experience, be very unusual to drink tea during the meal (see exception below). Rather than tea, people again usually drink beer or baijiu (Chinese rice wine). Those not wishing to drink alcohol usually drink Lilt, Coca Cola etc. (often the real stuff, sometimes local varieties - I'm guessing this is what others have seen on banquet tables.) Again, it may be difficult to get cold beer.
In recent years (2 or 3) grape wine - always red for some reason - has become popular at formal meals. It is usually locally produced and horrible. Even when it is good or, even better, imported, I absolutely hate the way they drink it. It is poured into tiny shot glasses and drunk as one of their endless 'toasts'. There is no chance to savour it - often a blessing. Alternatively, they pour the bottle into a jug and top it up with Lilt or 7-Up. Someone recently paid megadollars for a top Bordeaux and did just that - philistine!
One exception to the 'no tea with dinner' is, of course, morning tea or yum cha. This was traditionally a celebration of the tea (I'm sure most people know that 'yum cha' literally translates as 'drink tea'). The accompanying dim sum were to compliment the tea rather than vice versa. I'm sure that is less true today. People go for the grub, but tea is still usually served alongside.
In the family home, I find, as has been suggested above, that most people get their liquids from soup, although beer is again a favourite, 90% of the time among the men. Rice wine is still popular among the elderly - it is less fashionable than it once was. Just recently I've seen some of the women drinking this horribly sweet red 'tonic wine' - often home made. There is still a hangover of a stigma about women drinking.
For the record and to contribute to the post's main thrust, when I was a kid, I too was slowly introduced to wine with dinner (French mother). The idea of dinner without wine was inconceivable. We, too got it less and less watered down as we aged. When I'm back home in Europe, it is still always wine with dinner.
But, here in China, I tend to drink beer most days, but sometimes, usually weekends, I go crazy and have wine. Fortunately, because of my main job, I get crates of free good European, American etc wines delivered regularly. Haven't ran out in years!
The very idea of milk with dinner makes me cringe.
