My vague impression is that during the Early Republic, the USA was all about hard cider and whiskey. But through the 19th century, lots of different groups were coming over. The biggest, I'd think, were the Germans and their lagered beers, which I know eventually would come to be so dominant, but how quickly did something like that diffuse into the wider American population? What would a broad snapshot of American drinking habits have looked like in... 1840, 1870, 1900?
You're pretty much correct. Colonial American drinking habits were predominately British drinking habits taking advantage of local produce and grain: cider, whiskey, and rum. Rum was actually the biggest liquor until after the American Revolution, when depleting sugarcane fields in the Caribbean and British economic legislation made molasses more expensive and increasing grain farming in America made corn and rye cheap.
By 1840, this is pretty much how it was. Rum had mostly disappeared from the picture and whiskey was the main liquor from coast to coast. It was often raw product right out of the still (think a harsher version of the "moonshine" you find in glass jars as the liquor store), and a total lack of advertising and food safety laws meant it could be damn near anything. Your "Pure Kentucky Bourbon" would be crude neutral spirit made from molasses or corn, heavily watered down with lanolin, glycerin, prune juice, burnt sugar, acid, and other nasty stuff to give it the color, body, and burn of legitimate whiskey.
At this point, it's easy to do this because whiskey was not being sold by the distillery to the final consumer in most cases. It was first sold to wholesalers who would blend, rectify (redistill to remove impurities), and adulterate the spirit if they were scam artists. These wholesalers sold the whiskey under their brand name to the saloons and general stores that actually sold whiskey to customers.
By 1870, the biggest change is going to be in beer and mixed drinks. For beer, German immigrants arriving in America (especially after the potato blight of the 1840s hit Bavaria and other southern German regions) quickly began producing beer like they did in their homeland. Cities with a large German population like New York quickly sprouted up German lager saloons, though these were most frequented by Germans and served as community centers for immigrants. Beer did start to expand across the country, though mostly based on the German traditions rather than the larger variety present in Europe. Because American barley had a high protein content than led to hazy beers with off-flavors, these German brewers used rice or corn to lighten the body.
For mixed drinks, they've existed in some form since Ancient Rome and the "cocktail" (liquor, sugar, water/ice, bitters) dates back to at least the 1790s. After the Civil War, however, the US began seeing the importation of various fancy liqueurs and spirits like Chartreuse and Cointreau. Where cocktails were once simple creations, they now began to turn into elaborate concoctions of crushed ice and fruit juice, or pousse-cafes made of layered colored liqueurs. There was some pushback from the old guard who just wanted an old fashioned cocktail (see where the name "Old Fashioned" comes from?), but mixed drinks only became more and more elaborate.
The next change would be Prohibition. The rising cocktail culture suddenly had the door slammed in its face, which created a new drinking culture more suited to its illegal nature. Gin became the new spirit, as it was much easier for bootleggers to distill crude moonshine and flavor it to mask the impurities than it was to produce aged whiskey. Cocktails were made to further mask the flavor and be easy to quaff, like the Bee's Knees of gin, lemon juice, and honey. Rum, which had fallen from grace a century ago, saw a resurgence as travelers got their legal drinking done in the Caribbean and Bacardi of Cuba pioneered a heavily filtered light rum that was perfect for mixing.
Prohibition permanently changed American drinking habits. Many of the famous and talented bartenders of the 1910s had either stopped mixing altogether or fled to Europe and elsewhere to keep their jobs. The speakeasies had democratized drinking by providing a place for both men and women to go, which meant bartenders of the 1930s and on had a drive to cater to their newfound female clientele. This is the period when you start to see the Old Fashioned get muddled fruit and soda water added to it. Light, blended Canadian whiskey smuggled over the border created a taste for lighter tastes with dark liquor and gin and rum were now everywhere. The tiki craze began with Donn Beach and Victor Bergeron in the 1930s, creating complex rum-based cocktails with fruit juices and spices that hid massive amounts of alcohol behind sweet flavors.
The other change was beer: Prohibition had killed most of the breweries, leaving only the large companies that could produce other products to survive. These were the breweries by people like Busch, Schlitz, Miller, and Pabst, all immigrants from Germany who had been producing light adjunct lagers. Beer in America became a sort of standardized product with everyone drinking the exact same stuff. Every company advertised how different it was from the competition, but it all tasted mostly the same.
The Baby Boomer generation also rebelled against their parents. Seeing whiskey and cognac as the stuffy, foul-tasting drinks of the old guard, the youth who would come of age in the 1960s and 1970s switched to lighter, fruitier drinks using a new international spirit: vodka. Virtually tasteless and capable of being mixed with anything, vodka and light rum were the drinks of the new generation. Tequila was known in America before, but mainly near the Mexican border or by people who traveled to Mexico. Most of the tequila ending up in the US at this time was a terrible product, which is where the tradition of sucking a lime and a pinch of salt after a shot came from. Japanese restaurants sold sake, but it was often cheap stuff served hot to mask the taste.
The 1980s continued this by adding even more fruit juices and liqueurs, such as blue curacao (once a 19th century product made in small amounts) that gave an electric blue color and orange flavor to anything. The Long Island Iced Tea is sort of the nadir of this time, mixing every random booze you have behind the bar into a high-alcohol product that tastes completely generic and could be drank by a child without complaint.
The 1990s and 2000s are when things finally start turning around. The declining bourbon industry began to see a revitalization and even aged rum started returning to shelves. The discovery of old pre-Prohibition cocktail books and an emphasis on quality and precision by pioneering bartenders like the late Sasha Petrasky and Dale DeGroff saw a widening interest in good drinks with challenging flavors and unusual ingredients. The craft beer revolution, nascent since the 1965 purchase of the declining Anchor Brewing by Fritz Maytag III, was kicked into high gear by companies like Goose Island in Chicago.
The 2010s has been a time of fantastic change for liquor. After decades of decline in favor of instant gratification and simplicity, the American consumer is ready to get challenged again. Today you can walk into a liquor store in any major city and find Japanese whiskey costing $150 a bottle, Peychaud's bitters, dark beers brewed in Trappist monasteries, and tequila meant to be sipped like a fine scotch.