Although there are many fantastic bars in Venice, if you're planning a night time "pub crawl" you should plan a few places to visit in advance, otherwise it's very easy to waste an hour wandering aimlessly in search of a watering hole that's actually open (especially midweek).
There are two late-night drinking areas in Venice. Piazza San Marco is NOT one of them. Although it is very pleasant and there are many people wandering around late. But the actual late night scene is in either Campo Santa Margherita, near the University Ca' Foscari in Dorsoduro; or in Erbaria on the West side of the Rialto Bridge where the main vegetable market is held during the day.
Try a Spritz (with either Campari, Select or Aperol), a typical drink loved by all Venetians that's usually drunk while eating cicheti. You can find it in almost every bar in the city. Price is about €2, more in a touristy place.
If you try the famous Veneto Grappa be careful--it's almost pure alcohol.
The Bellini was invented in Harry's Bar in Venice. It is a mix of white peach juice and Prosecco (the ubiquitous Venetian Champagne-like sparkling wine). Fermented at a low temperature Prosecco develops amylic aromas (fruit drops), though these perhaps mix better with fruit juices than does the more austere Champagne. Classic Bellinis should never be made with Champagne. Although by normal standards expensive, a Bellini in Harry's Bar (€17 for a 1.5 oz drink is obscene) is still much cheaper than on the terraces of similar '5-star' establishments in the city.
Beer in a small pub is about €5 for a pint (birra media).
Espresso, the real italian, is about €1 at the bar, €2 at a table.
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Based on a work at Wikitravel.org & Traveldudes.org.