Yeah, we "go to the beach" (and swim in ocean) and if using past tense, i would say "i was at the beach yesterday" (i might go on and say "the water was cold, (or warm, or rough, or..)--since most often, one goes to the beach to swim.

If i went to a lake to swim, i would say 'We went to lake welsch" (harriman state park) or some other lake. (and most would presume for swimming,--if you went out rowing on the lake or fishing, you'd say that) (and i would presume the same if Jackie or anyone else said "we went to the lake (or to lake such and such)"

no one swims in North(hudson river) in NY metro area.
the current is that strong, but there are no safe area's to enter the water (not to mention, its not all that clean--it much cleaner that it was 20 years ago, but...) NY's other 'rivers' aren't rivers (harlem river, east river) but tidal straights (and in the 'east river' the tides cause a current that can 'run' as fast as 20 knots--so its definately not safe to swim in! (these salt water 'rivers' never freeze)

there are some beaches in NYC on Long Island Sound (in the bronx) most of NYC's beaches are ocean beaches --coney island-brooklyn) and Rockaway--queens are the big ones, there are others (really extention of the same barrier islands, just different sections have specific names. (west end of rockaway is 'breezy point', east end is "far rockaway" --but an out of towner would have trouble recognizing these as different 'beaches' (one barrier island, a 10 miles stretch of sandy shore!)

as for rinks, there is at least one ice skating rink per borough, (free or low cost city park rinks) and i know in many parts of US people build out door rinks by just setting up some 2X4's and filling them with water and letting them freeze. In NYC is rarely cold enought to make such a rink worthwhile--though i suspect some upstater's might do it.
the netherlands, with its canals and northern climate, is unique in having 'winter/ice' highways for skating.